Question

Find the area between the curve
y
=
x
3
, the axis of
y
and the lines
y
=
1
and
y
=
8

Answer 1

Answer:

  (d)  11 1/4 units²

Step-by-step explanation:

You want the area bounded by the x-axis, the lines y=1 and y=8, and the curve y=x^3.

Area

Since the y-axis is one bound it works well to use a differential of area that is (x2 -x1)dy, where x1 = 0, and x2 = ∛y. The limits of the integration will be the boundary lines y=1 and y=8. The power rule is used for integration.

  $\displaystyle A=\int_{1}^{8}{y^{\frac{1}{3}}}\,dy=\left.\dfrac{3}{4}y^{\frac{4}{3}}\right|_{1}^8=\dfrac{3}{4}(16-1)=\dfrac{45}{4}=\boxed{11\dfrac{1}{4}\text{ units}^2}$

__

Additional comment

The given curve is y=x³. Solving for x, we find x=∛y. This is the boundary curve we used for integration.

The attachment also shows the integral over x. For that, the region is divided into two parts: a rectangle to the left of x=1, and the region bounded on the bottom by y=x³ to the right of x=1. The x-value corresponding to y=8 is x=2, so that is the limit of integration for the part of the region to the right of x=1.