Question

The boundaries of the shaded region are the y-axis, the line y=1, and the curve y=x^(1/4). Find the area of this region by writing x as a function of y and integrating with respect to y.

Answer 1

To find the area of the shaded region, you can write x as a function of y and integrate with respect to y. The area is equal to the integral of y^(4) from 1 to 0. The area of the shaded region is 1/5 square units.

Step-by-step explanation:

To find the area of the shaded region, we need to determine the upper and lower bounds of integration. The lower bound is the y-coordinate of the line y = 1, and the upper bound is the y-coordinate of the curve y = x^(1/4). We can rewrite the equation y = x^(1/4) as x = y^(4).

Now, we can set up the integral to find the area:

Area = ∫1y^(4)(1 - 0) dy

Simplifying, we get:

Area = ∫1y^(4) dy

Integrating with respect to y, we get:

Area = (1/5)y^(5) + C

Since the area is bounded by y = 1 and the y-axis, we substitute y = 1 into the formula:

Area = (1/5)(1)^(5) + C = 1/5

Therefore, the area of the shaded region is 1/5 square units.