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6.1 pt 2 Area of a Region Between Two Curves

6.1 pt 2 Area of a Region Between Two Curves. HW pg 413 22-40, 52, 60 (even). The formula for the area between curves is:. We will use this so much, that you won’t need to “memorize” the formula!. AREAS BETWEEN CURVES.

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6.1 pt 2 Area of a Region Between Two Curves

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  1. 6.1 pt 2 Area of a Region Between Two Curves HW pg 413 22-40, 52, 60 (even)

  2. The formula for the area between curves is: We will use this so much, that you won’t need to “memorize” the formula!

  3. AREAS BETWEEN CURVES • To find the area between the curves y = f(x) and y = g(x), where f(x) ≥ g(x) for some values of x but g(x)≥ f(x)for other values of x, split the given region S into several regions S1, S2, . . . with areas A1, A2, . . .

  4. AREAS BETWEEN CURVES • Then, we define the area of the region Sto be the sum of the areas of the smaller regions S1, S2, . . . , that is, A = A1 + A2 + . . .

  5. AREAS BETWEEN CURVES • Since • we have the following expression for A.

  6. AREAS BETWEEN CURVES Definition 3 • The area between the curves y = f(x) and y = g(x) and between x = a and x = b is: • However, when evaluating the integral, we must still split it into integrals corresponding to A1, A2, . . . .

  7. AREAS BETWEEN CURVES Example 5 • Find the area of the region bounded by the curves y = sin x, y = cos x,x =0, and x = π/2.

  8. AREAS BETWEEN CURVES Example 5 • The points of intersection occur when sin x = cos x, that is, when x = π/ 4 (since 0 ≤ x ≤ π/2).

  9. AREAS BETWEEN CURVES Example 5 • Observe that cos x≥ sin x when 0 ≤ x ≤ π/4 but sin x≥ cos x when π/4≤ x ≤ π/2.

  10. AREAS BETWEEN CURVES Example 5 • So, the required area is:

  11. AREAS BETWEEN CURVES Example 5 • We could have saved some work by noticing that the region is symmetric about x=π/4. • So,

  12. AREAS BETWEEN CURVES • Some regions are best treated by regarding x as a function of y. • If a region is bounded by curves with equations x = f(y), x = g(y), y = c, and y = d, where f and gare continuous and f(y) ≥ g(y) for c≤ y ≤ d, then its area is:

  13. AREAS BETWEEN CURVES • If we write xRfor the right boundary and xL for the left boundary, we have: • Here, a typical approximating rectangle has dimensions xR - xLand ∆y.

  14. AREAS BETWEEN CURVES Example 6 • Find the area enclosed by the line y = x - 1 and the parabola y2 = 2x + 6.

  15. AREAS BETWEEN CURVES Example 6 • By solving the two equations, we find that the points of intersection are (-1, -2) and (5, 4). • We solve the equation of the parabola for x. • From the figure, we notice that the left and right boundary curves are:

  16. AREAS BETWEEN CURVES Example 6 • We must integrate between the appropriate y-values, y =-2 and y =4.

  17. AREAS BETWEEN CURVES Example 6 • Thus,

  18. AREAS BETWEEN CURVES • In the example, we could have found the area by integrating with respect to x instead of y. • However, the calculation is much more involved.

  19. AREAS BETWEEN CURVES • It would have meant splitting the region in two and computing the areas labeled A1 and A2. • The method used in the example is much easier.

  20. If we try vertical strips, we have to integrate in two parts: We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.

  21. We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y. length of strip width of strip

  22. 1 General Strategy for Area Between Curves: Sketch the curves. Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.) 2 3 Write an expression for the area of the strip. (If the width is dx, the length must be in terms of x. If the width is dy, the length must be in terms of y. 4 Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.) 5 Integrate to find area. p

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