Question

Sketch the region enclosed by

y
=
5
x
and
y
=
8
x
2
. Find the area of the region.

Answer 1

125/384 ≈ 0.32552

Explanation:

You want the area between the curves y = 5x and y = 8x².

Area

The difference between the curves is ...

f(x) = 5x -8x² = x(5 -8x)

This difference is zero when ...

x = 0

5 -8x = 0 ⇒ x = 5/8

The area will be the integral of f(x) with the limits 0 and 5/8:

`\displaystyle \text{area}=\int_0^(5)/(8){(5x-8x^2)}\,dx=(5)/(2)\cdot\left((5)/(8)\right)^2-(8)/(3)\cdot\left((5)/(8)\right)^3\\\\\\\text{area}=\left((5)/(8)\right)^2\left((5)/(2)-(8)/(3)\cdot(5)/(8)\right)=(25)/(64)\cdot(5)/(6)=\boxed{(125)/(384)}`

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