8.5 Finding the Area Between Curves Expressed as Functions of y

Think in horizontal slices instead of vertical ones. If your curves are x = f(y) (or you can invert y = g(x) to get x in terms of y), draw the region and slice with a thin horizontal strip of height dy. Each strip has length (x_right − x_left), so the area is A = ∫[y1 to y2] (x_right(y) − x_left(y)) dy. Steps: - Sketch and label which curve is right and which is left for each y. - Find y-bounds y1 and y2 from intersections (solve for y-values). - Write x_right(y) and x_left(y) (invert if needed). - Integrate A = ∫_{y1}^{y2} [x_right(y) − x_left(y)] dy. - If the region changes which curve is right/left, split into pieces and add integrals. This is exactly what CED CHA-5.A describes (use either x or y). For a refresher with examples, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and more Unit 8 review (https://library.fiveable.me/ap-calculus/unit-8). Practice lots of problems from Fiveable’s problem set (https://library.fiveable.me/practice/ap-calculus) so you’re ready for AP-style questions.

If you set up horizontal slices (integrate with respect to y), the area of a region between two curves x = f_right(y) and x = f_left(y) from y = a to y = b is A = ∫[y=a to b] (x_right(y) − x_left(y)) dy Key steps: solve for x as functions of y (invert if needed), sketch and identify the rightmost and leftmost curves for each y, find the y-bounds a and b from intersection points (solve f_right(y)=f_left(y)), and watch for piecewise changes in which curve is rightmost. Differential area element: dA = (x_right − x_left) dy. This is the same AP CHA-5.A idea as using x-slices, just rotated—practice converting functions and finding y-intersections. For the AP exam, show a clear sketch, state your a and b, and label which is right/left before integrating. See the Topic 8.5 study guide for examples and strategies (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar). For more practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

Use dy when slicing the region with horizontal strips (thin horizontal rectangles). That happens when it’s easier to express the boundary curves as x = f(y) (or to invert y = g(x) to get x in terms of y), or when the left/right edges of the region are naturally given by functions of y. Then dA = (x_right(y) − x_left(y)) dy and your integral runs between the y-values of the intersection points (find y-bounds). Quick checklist: - If the region is bounded left/right by curves x = f(y), use dy. - If the region is bounded top/bottom by y = f(x), use dx. - If inverting is messy or the region is better sliced horizontally (e.g., a sideways parabola), choose dy. - Always sketch, find intersection y-values, and write integrand as right − left. This is exactly the CHA-5.A idea in the CED: areas can be calculated with respect to x or y using horizontal strips and x = f(y) (see the topic study guide for examples: https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar). For extra practice, check Unit 8 overview (https://library.fiveable.me/ap-calculus/unit-8) and plenty of problems at (https://library.fiveable.me/practice/ap-calculus).

Think of slicing the region: integrating with respect to x uses vertical slices (dA = (top − bottom) dx), while integrating with respect to y uses horizontal slices (dA = (right − left) dy). Use dx when the region is easiest described as y = f(x) with clear x-bounds; use dy when the region is easier as x = g(y) with clear y-bounds. Key steps for dy: solve/invert the curves to get x = f(y), find the intersection y-values (y-bounds), sketch horizontal strips, and set Area = ∫[y1 to y2] (x_right(y) − x_left(y)) dy. Sometimes a region needs piecewise integrals in y if which curve is left/right changes. Both approaches are allowed on the AP (CED CHA-5.A.2); pick the one that gives a single simple integral. For a quick refresher see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Think horizontal: slice the region with thin horizontal strips (dA = (x_right − x_left) dy). Steps: 1. Sketch the curves x = f(y) and x = g(y) and solve f(y) = g(y) to get the y-bounds y1 and y2 (intersection y-values). 2. For each y between y1 and y2 identify the rightmost x-value (call it x_right) and the leftmost x-value (x_left). Often x_right = max{f(y), g(y)}. 3. Set up the area: A = ∫_{y1}^{y2} [x_right(y) − x_left(y)] dy. 4. Evaluate. If one curve isn’t written as x = something, invert y = h(x) to x = h^{-1}(y) on the relevant interval or split into pieces where the left/right roles change. This matches the CED CHA-5.A idea: integrate with respect to y using horizontal strips; dA = (x_right − x_left) dy. For more examples and practice, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and the Unit 8 overview (https://library.fiveable.me/ap-calculus/unit-8). For extra practice problems, try the AP practice bank (https://library.fiveable.me/practice/ap-calculus).

Think in terms of horizontal slices (dA = (x_right − x_left) dy). If you’re integrating with respect to y, solve each curve for x = f(y) and then subtract the leftmost x from the rightmost x on each horizontal strip. Steps you can use every time: - Sketch the region and draw a horizontal slice. - Solve the curves for x in terms of y (invert if needed). - Find the y-bounds by solving intersections (y-values). - On each y-interval, identify which curve gives the larger x (rightmost) and which gives the smaller x (leftmost). Integrand = x_right(y) − x_left(y). - If the “right” and “left” switch, split the integral at that y and do piecewise integrals. This is exactly the CHA-5.A idea: integrate with respect to y, use horizontal strips, and compute rightmost minus leftmost. For more examples and guided practice, check the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and try problems on Fiveable’s practice page (https://library.fiveable.me/practice/ap-calculus).

Start by sketching the region and draw horizontal slices (since you’ll integrate with respect to y). Step-by-step: 1. Identify the curves and rewrite them as x = f(y) (invert y = g(x) if needed). You need x expressed in terms of y so each horizontal slice has left and right x-values. 2. Find the y-bounds: solve intersection points of the curves to get y_min and y_max (use the y-values of intersections). 3. For a typical horizontal strip, dA = (x_right(y) − x_left(y)) dy. Choose the rightmost and leftmost x for each y in the interval. 4. Set up the definite integral: Area = ∫[y_min to y_max] (x_right(y) − x_left(y)) dy. If the left/right roles change, split into pieces and sum integrals. 5. Evaluate the integral(s). Key AP CED ideas: integrate with respect to y using horizontal strips, use x = f(y), and watch for piecewise boundaries where you must split the integral. For a short walkthrough and examples, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar). For broader review and lots of practice problems, check Unit 8 (https://library.fiveable.me/ap-calculus/unit-8) and the practice bank (https://library.fiveable.me/practice/ap-calculus).

Find where the curves meet by working in y: write both boundaries as x = f(y) and x = g(y), then solve f(y) = g(y) for y. Those solutions are your y-bounds (intersection y-values). Steps: - If given y = h1(x) and y = h2(x), invert to x = h1^{-1}(y) and x = h2^{-1}(y) when possible. - Set x_right(y) = x_left(y) and solve algebraically for y (or use a graph/calculator). - Sketch and check which curve is right vs left for each y-interval (horizontal strips). - Area = ∫_{y=a}^{y=b} [x_right(y) − x_left(y)] dy; use multiple integrals if the left/right order changes. Example quick check: if curves are y = x^2 and x = y − 2 (i.e. x = y − 2 and x = ±√y), set y − 2 = √y (or = −√y) to find intersection y-values, then integrate with respect to y. For AP CED language: invert functions, use horizontal slices, dA = (x_right − x_left) dy, and be explicit about y-bounds. For more examples and practice, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar), the Unit 8 overview (https://library.fiveable.me/ap-calculus/unit-8), and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Use functions of y when horizontal slices make the setup simpler or even possible. If the region is naturally bounded left/right by curves x = f(y) (or when solving for x in terms of y is easier than inverting y = f(x)), integrate with respect to y using area = ∫[y1 to y2] (x_right(y) − x_left(y)) dy. The CED explicitly allows areas computed with either variable (CHA-5.A.2). Quick rules of thumb: - Sketch the region and try both vertical and horizontal slices. Pick the one that gives “right − left” (for dy) or “top − bottom” (for dx) as a single simple expression. - If curves intersect at y-values that are easy to find but their x-intersections are messy, use dy. - If the region changes which curve is on top/left over the interval, you may need piecewise integrals—choose the variable that minimizes pieces. For worked examples and AP-style practice using functions of y, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and Unit 8 overview (https://library.fiveable.me/ap-calculus/unit-8). For lots of extra practice, try the practice bank (https://library.fiveable.me/practice/ap-calculus).

When you integrate with respect to y, your limits are the y-values that bound the region: the lower y (a) and upper y (b). To find them, solve for the intersection points of the boundary curves and read off their y-coordinates (these are your a and b). Then slice the region with horizontal strips: each strip’s area = (x_right(y) − x_left(y)) dy, so the area = ∫[a to b] (x_right − x_left) dy. If the region changes which curve is left/right for some y, split the integral at the y where that switch happens (use piecewise integrals). Also remember to rewrite curves as x = f(y) (invert if needed) before integrating. This is exactly the CHA-5.A idea in the CED. For a worked refresher, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Think of slicing the region with horizontal strips instead of vertical ones. For functions given as x = f(y), take a thin horizontal strip at height y with thickness dy. Each strip’s area ≈ (x_right(y) − x_left(y))·dy, so the total area is ∫[y1 to y2] (x_right − x_left) dy. Steps to visualize and set up the integral: - Sketch both curves and draw horizontal slices (CED keywords: horizontal strips, sketch with horizontal slices). - Solve each curve for x in terms of y (invert if needed) so you have x = f(y). - Find y-bounds by intersecting the curves and get the y-values y1, y2. - Identify which curve is rightmost and which is leftmost on [y1,y2]. - Compute A = ∫_{y1}^{y2} (x_right(y) − x_left(y)) dy. Tip: if inversion is messy, check if integrating with respect to x is simpler. For more examples and AP-aligned practice, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar), the Unit 8 overview (https://library.fiveable.me/ap-calculus/unit-8), and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Think in horizontal slices: when you integrate with respect to y you use dA = (x_right(y) − x_left(y)) dy. Steps to know which is which: - Sketch the region and draw a horizontal line at a typical y. The curve with the larger x-value on that horizontal line is x_right(y); the smaller x-value is x_left(y). - Solve each curve for x as a function of y (invert y = f(x) if needed) so you have x = g1(y), x = g2(y). - Find the y-bounds from intersection points (solve for y where the two curves meet). - On the interval of y between intersections, evaluate a sample y to see which x-function is larger (or just compare expressions algebraically). - If the region changes left/right ordering, split the integral at the y where ordering switches and add two integrals. This is exactly the “rightmost minus leftmost” idea in the CED (CHA-5.A.2): use horizontal strips, invert functions to x = f(y), and integrate between y-bounds. For a refresher and examples, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and unit review (https://library.fiveable.me/ap-calculus/unit-8). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

Sketch horizontal slices (or solve for intersections) first: set y^2 = 2y − y^2 ⇒ 2y^2 − 2y = 0 ⇒ y( y − 1) = 0 so y = 0 and y = 1. Use horizontal strips: dA = (x_right − x_left) dy. For a test y (e.g., y = 0.5) 2y − y^2 = 0.75 > y^2 = 0.25, so the right curve is x = 2y − y^2 and the left is x = y^2. Set up the integral: A = ∫_{y=0}^{1} [ (2y − y^2) − (y^2) ] dy = ∫_{0}^{1} (2y − 2y^2) dy. You can evaluate it if you need: A = [y^2 − (2/3) y^3]_{0}^{1} = 1 − 2/3 = 1/3. This follows the CED rule: integrate with respect to y, rightmost minus leftmost, using y-bounds from intersection points. For extra practice and similar examples see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Switch to integrating with respect to y whenever horizontal slices make the region easier to describe. Concretely: if the region is bounded left/right by functions x = f(y) (or when solving for x in terms of y is simple), or if a horizontal strip gives a single “rightmost − leftmost” expression, integrate dy. Steps: sketch the region, find the y-bounds by solving intersections for y, write dA = (x_right − x_left) dy, and set up ∫[y1 to y2] (x_right(y) − x_left(y)) dy. Use dy when vertical strips would require splitting into pieces or inverting messy functions. Remember to invert functions correctly (or solve for x) and watch for piecewise boundaries in y. AP tip: the CED expects you to calc area using either variable (CHA-5.A.2); include correct y-bounds and justify your choice on free-response parts (Topic 8.5). For a focused walk-through, see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar). For extra practice, try problems from Unit 8 (https://library.fiveable.me/ap-calculus/unit-8) or the practice bank (https://library.fiveable.me/practice/ap-calculus).

Short answer: when you integrate with respect to y, always do (right function − left function). Top − bottom is for dx integrals (w.r.t. x). Why: dA = (x_right − x_left) dy (horizontal strips). So rewrite the curves as x = f(y) (invert if needed), sketch horizontal slices, find which curve gives the larger x at a typical y (that’s the rightmost). Use the intersection y-values as your bounds. Quick example: region bounded by x = y^2 and x = 2y. Intersections: y=0 and y=2. Area = ∫[0 to 2] (2y − y^2) dy. Tips: always sketch and draw a horizontal slice, label x_right and x_left, and solve for intersections in y. If a region needs piecewise setup in y, break the integral accordingly. For more practice and CED-aligned guidance see the Topic 8.5 study guide (https://library.fiveable.me/ap-calculus/unit-8/finding-area-between-curves-expressed-as-functions-y/study-guide/PHhLi2WAxSUrfKfoDQar) and try problems on the Unit 8 practice page (https://library.fiveable.me/practice/ap-calculus).