Use an iterated integral to find the area of the region bounded by the graphs of the equations

Question

Quizplus · Accepted Answer

To find the area of the region bounded by the given curves, we need to integrate the expression 11ebbd4d_3f0c_3d26_8a0a_335a3e1499b2_TB8202_11 - 11ebbd4d_3f0c_3d27_8a0a_37df44998ede_TB8202_11 over the given interval. We can set up the iterated integral as follows:

∫[0,2]∫[x^2,2x] (11ebbd4d_3f0c_3d26_8a0a_335a3e1499b2_TB8202_11 - 11ebbd4d_3f0c_3d27_8a0a_37df44998ede_TB8202_11) dy dx

Simplifying the limits of integration by solving the equation x^2 = 2x for x, we get x = 0 and x = 2. Therefore, the iterated integral becomes:

∫[0,2]∫[x^2,2x] (11ebbd4d_3f0c_3d26_8a0a_335a3e1499b2_TB8202_11 - 11ebbd4d_3f0c_3d27_8a0a_37df44998ede_TB8202_11) dy dx
= ∫[0,2] (11ebbd4d_3f0c_3d26_8a0a_335a3e1499b2_TB8202_11y - 11ebbd4d_3f0c_3d27_8a0a_37df44998ede_TB8202_11y^2/2)[y=x^2 to y=2x] dx
= ∫[0,2] [(11ebbd4d_3f0c_3d26_8a0a_335a3e1499b2_TB8202_11*2x - 11ebbd4d_3f0c_3d27_8a0a_37df44998ede_TB8202_11*(2x)^2/2) - (11ebbd4d_3f0c_3d26_8a0a_335a3e1499b2_TB8202_11x^2 - 11ebbd4d_3f0c_3d27_8a0a_37df44998ede_TB8202_11*x^4/2)] dx
= ∫[0,2] (-7x^2 + 22x) dx
= (-7/3)x^3 + 11x^2 |[0,2]
= [(16/3)*2^3 + (44/3)*2] - 0
= (64/3 + 88/3)
= 152/3

Thus, the area of the region bounded by the given curves is 152/3 square units, which corresponds to choice (B).

Use an Iterated Integral to Find the Area of the Region