Question

Find the area bounded by y = 5 + 4x − x2, the x-axis and the ordinates x = 1 and x = 4.

a. 24 unit^2
b. 19 unit^2
c. 2 unit^2
d. 32 unit^2

Answer 1

Question #31154

Find the area bounded by $\gamma = 5 + 4x - x^2$, the x-axis and the ordinates $x = 1$ and $x = 4$.

a. 24 unit^2

b. 19 unit^2

c. 2 unit^2

d. 32 unit^2

Solution. The area bounded by the function $y = 5 + 4x - x^2$, x-axis and $x = 1$ and $x = 4$ is equal to integral

$$\begin{array}{l} S = \int_{1}^{4} (5 + 4x - x^2) dx = \left(5x + 2x^2 - \frac{x^3}{3}\right) \int_{1}^{4} = 5 \cdot 4 + 2 \cdot 4^2 - \frac{4^3}{3} - 5 \cdot 1 - 2 \cdot 1^3 + \frac{1}{3} \\ = 20 + 32 - \frac{64}{3} - 7 + \frac{1}{3} = 45 - \frac{63}{3} = \frac{72}{3} = 24. \end{array}$$

Answer. The area is $S = 24 \text{ unit}^2$.