Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      
                                                                                                                                                                                       
 ∫₀⁴ (8―2𝓍) d𝓍

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).

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 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

{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of ƒ and the 𝓍-axis on the given interval. Graph ƒ and show the region 𝑅.                                              

 ƒ(𝓍) = 2 ― |𝓍| on [ ― 2 , 4]

Find the area of the shaded regions in the following figures.

Determine the area of the shaded region bounded by the curve x^2=y^4(1−y^3) (see figure).

Find the area of the following regions, expressing your results in terms of the positive integer n≥2.

Let Aₙ be the area of the region bounded by f(x)=x^1/n and g(x)=x^n on the interval [0,1], where n is a positive integer. Evaluate lim n→∞ Aₙ and interpret the result. br