Find the definite integral by computing an area.
-$$\int _ { 0 } ^ { 8 } ( k x + 3 ) d x ; k \geq 0$$
A) $32 k + 24$
B) $16 k - 24$
C) $64 k + 24$
D) $16 \mathrm { k } + 24$

Question

Quizplus · Accepted Answer

The integral $\int _ { 0 } ^ { 8 } ( k x + 3 ) d x$ represents the area under the curve $y = kx + 3$ from $x = 0$ to $x = 8$. To find this area, we can compute the integral directly:$$\begin{align*}\int _ { 0 } ^ { 8 } ( k x + 3 ) d x &= \left[ \frac{1}{2} k x^2 + 3x \right]_0^8 \&= \left( \frac{1}{2} k (8)^2 + 3(8) \right) - \left( \frac{1}{2} k (0)^2 + 3(0) \right) \&= 32k + 24.\end{align*}$$

Find the Definite Integral by Computing an Area ∫08(kx+3)dx;k≥0\int _ { 0 } ^ { 8 } ( k x + 3 ) d x ; k \geq 0∫08​(kx+3)dx;k≥0

Find the Definite Integral by Computing an Area $\int _ { 0 } ^ { 8 } ( k x + 3 ) d x ; k \geq 0$