Question

Determine the area bounded by the curve y=3x2 + 6x + 8, the x-axis and the ordinates x=1 and x =3.

a. 6 unit^2
b. 6 unit^2
c. 66 unit^2
d. 65 unit^2

Answer 1

Determine the area bounded by the curve $y = 3x^{2} + 6x + 8$, the x-axis and the ordinates $x = 1$ and $x = 3$.

a. 6 unit²

b. 6 unit²

c. 66 unit²

d. 65 unit²

**Solution:**

For the area under the curve we have formula:

$$A = \int_{a}^{b} y \, dx$$

In our case we have:

$$\begin{array}{l} A = \int_{1}^{3} (3x^{2} + 6x + 8) \, dx = \left(3\frac{x^{3}}{3} + 6\frac{x^{2}}{2} + 8x\right) \|_{1}^{3} = (x^{3} + 3x^{2} + 8x) \|_{1}^{3} \\ = 3^{3} + 3 * 3^{2} + 8 * 3 - 1^{3} - 3 * 1^{2} - 8 * 1 = 27 + 27 + 24 - 1 - 3 - 8 = 66 \end{array}$$

Answer: c. 66 unit²