9. Graphical Applications of Integrals

Determine the area of the shaded region in the following figures.

Determine the area of the shaded region in the following figures.

Determine the area of the shaded region in the following figures.

Determine the area of the shaded region in the following figures.

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(a) A (―2)

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(4)

For the given regions R₁ and R₂, complete the following steps.

a. Find the area of region R₁.

R₁ is the region in the first quadrant bounded by the y-axis and the curves y=2x^2 and y=3−x; R₂ is the region in the first quadrant bounded by the x-axis and the curves y=2x^2 and y=3−x(see figure).

For the given regions R₁ and R₂, complete the following steps.

a. Find the area of region R₁.

R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.

For the given regions R₁ and R₂, complete the following steps.

b. Find the area of region R₂ using geometry and the answer to part (a).

R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.

Find the area of the region described in the following exercises.

The region bounded by x=y(y−1) and y=x/3

Find the area of the region described in the following exercises.

The region in the first quadrant bounded by y=5/2−1/x and y=x

Determine whether the following statements are true and give an explanation or counterexample.

a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.

Determine whether the following statements are true and give an explanation or counterexample.

b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

Find the area of the region described in the following exercises.

The region bounded by y=x^2−2x+1 and y=5x−9

Determine whether the following statements are true and give an explanation or counterexample.

c. ∫₀¹(x−x^2) dx=∫₀¹(√y−y) dy