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.

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.

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

Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of ƒ(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and 𝓍= 6