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7.2 Areas Between Curves

7.2 Areas Between Curves. Area. Region R is bounded by the curves y = 2 – x 2 and y = -x . Sketch region R . R. What is the area of region R ?. Process. To find the area between curves: Sketch the region defined in the problem.

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7.2 Areas Between Curves

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  1. 7.2 Areas Between Curves

  2. Area • Region Ris bounded by the curves y = 2 – x2 and y = -x. • Sketch region R. R • What is the area of region R?

  3. Process • To find the area between curves: • Sketch the region defined in the problem. • Connect the curves with either a vertical strip (dx) or a horizontal strip (dy). • A strip that always connects the two curves will allow you to find the area without breaking up integrals. • Write an expression for the length of the rectangular strips. • Vertical Strips: Length = Top curve – Bottom curve • Horizontal Strips: Length = Right curve – Left curve • NOTE: IF YOU USE A dy STRIP, YOU MUST SOLVE THE CURVE FOR x IN TERMS OF y. • Add rectangular strips together by setting up an integral using your expression. • Find points of intersection. • NOTE: If using a dx, use the x-coordinates of intersection. If using a dy, use the y-coordinates of intersection.

  4. Area y = 2 – x2 y = -x dx Intersection 2 – x2 = –x –x2 + x + 2 = 0 –1(x2– x – 2) = 0 –1(x + 1)(x – 2) = 0 x = –1 x = 2 • What is the area of region R? Using a dx strip because it always connects the two curves. Length of dx strip = top – bottom = (2 – x2) – (-x) Bounds??? = –x2 + x + 2

  5. Example • Find the area bounded by y = ex, y = e2, and the y-axis. Strip? Length? dx Bounds? x = 0 and intersection (e2 = ex x = 2)

  6. Example • Find the area between the two curves x = y2 – 4y and y = x bounded by the x-axis. Strip? Length? Right – Left dy Bounds? y = 0 and intersection (y2 – 4y = y y = 5)

  7. Homework • Section 7.2 (#1-25 odd, 27-42 multiples of 3, 48)

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