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Between a line and a cubic

CalculusEasy

Find the area of the region in the first quadrant enclosed by the line $y = 4 x$ and the curve $y = x^{3}$ . Give the answer as an integer.

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Hint

Find where they meet, then integrate the upper function minus the lower one.

Solution

The graphs meet where $4 x = x^{3}$ ; in the first quadrant that is at $x = 0$ and $x = 2$ .
Between them the line lies above the curve, since at $x = 1$ we have $4 > 1$ .
The area is the integral of $4 x - x^{3}$ from $0$ to $2$ , which is the change in $2 x^{2} - \frac{x ^{4}}{4}$ between the limits.
That gives $8 - \frac{16}{4} = 4$ .

Answer: $4$