Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 0 } ^ { 5 } ( 6 t + 1 ) d t$
A) $- 158$
B) 12
C) 80
D) 82
E) $- 160$

Question

Quizplus · Accepted Answer

The given definite integral represents the area under the curve of the function $f(t) = 6t+1$ from $0$ to $5$.

To sketch the region, we can start by plotting the $x$- and $y$-axes and identifying the $t$-interval $[0,5]$. Next, we can plot the line $y = f(t) = 6t+1$ and shade the region underneath it and above the $x$-axis over the interval $[0,5]$. This region is a trapezoid with height $f(0) = 1$ and $f(5) = 31$, and bases of length $5$ and $0$, respectively.

We can find the area of this trapezoid using the formula for the area of a trapezoid:
$A = \frac{1}{2}h(b_1+b_2)$

where $h$ is the height (i.e., the distance between the bases) and $b_1$ and $b_2$ are the lengths of the bases. Substituting the given values, we get:
$A = \frac{1}{2}(30)(5+0) = 75(1) = 75$

Therefore, the answer is $A = \boxed{	extbf{(C)}\ 80}$.

Sketch the Region Whose Area Is Given by the Definite ∫05(6t+1)dt\int _ { 0 } ^ { 5 } ( 6 t + 1 ) d t∫05​(6t+1)dt

Sketch the Region Whose Area Is Given by the Definite $\int _ { 0 } ^ { 5 } ( 6 t + 1 ) d t$