Find the areas of the regions enclosed by the curves and lines...

Question

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.

x = 2y², x = 0, y = 3

Accepted Answer

Welcome back everyone. Find the area of the shaded region A 1 square unit, B 3 square units, C 1/2 square units, and D 2 square units. For this problem, the shaded region is bounded by the y axis. A horizontal line in the form of y equals 1. And we also have a curve in the form of x equals 3y2. So what we can do is simply remember that we can calculate the area as the integral from a to b of F of Y minus g of y. With respect toy, essentially f of y is the right curve. In this case, notice that our right curve is the blue one, which is x equals 3 y 2. So this is going to be f of y. Our left curve is going to be the y axis. Remember that y axis can be written as x equals 0, right? So this is going to be our g of y. And for the limits of integration, notice that in terms of the y axis, the region extends from 0 up to 1. So we can write area equals integral from 0 to 1. F of Y is the right curve, so we're going to take 3 Y 2d and we're going to subtract G of Y, which is 0. Let's go ahead and integrate. This is going to be equal to 3 integral from 0 to 1. Of y2 d y, here we can apply the power rules to get 3 multiplied by y cubed divided by 3. Evaluated from 0 to 1, which is y cubed. From 0 to 1. And this is going to be equal to 1 cubed minus 0 cubed, which is basically equal to 1. So the correct answer is option A 1 square unit. Thank you for watching.

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.x = 2y², x = 0, y = 3

Watch next

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
x = 2y², x = 0, y = 3