Use the following property of the definite integral to estimate the definite integral $\int _ { z / 6 } ^ { \pi / 4 } 6 \sin x d x$ :
If $m \leq f ( x ) \leq M$ on $[ a , b ]$, then
$$m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$$
A)
$\frac { \pi } { 2 } \leq \int _ { s / 6 } ^ { x / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 2 }$
B) $\frac { \sqrt { 2 } \pi } { 4 } \leq \int _ { z / 6 } ^ { s / 4 } 6 \sin x d x \leq \frac { \pi } { 4 }$
C)
$\frac { \pi } { 4 } \leq \int _ { s / 6 } ^ { s / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 4 }$
D) $\frac { \sqrt { 2 } \pi } { 2 } \leq \int _ { s / 6 } ^ { \pi / 4 } 6 \sin x d x \leq \frac { \pi } { 2 }$

Question

Quizplus · Accepted Answer

The function $6 \sin x$ is bounded by $0$ and $6$ on the interval $[z/6, \pi/4]$. The maximum value of $\sin x$ is $1$, so $6 \sin x$ is bounded by $0$ and $6$. The length of the interval is $\pi/4 - z/6$. Therefore, the integral is bounded by $0 \times (\pi/4 - z/6)$ and $6 \times (\pi/4 - z/6)$. The correct choice is C, which provides a reasonable estimate within these bounds.

Use the Following Property of the Definite Integral to Estimate ∫z/6π/46sin⁡xdx\int _ { z / 6 } ^ { \pi / 4 } 6 \sin x d x∫z/6π/4​6sinxdx

Use the Following Property of the Definite Integral to Estimate $\int _ { z / 6 } ^ { \pi / 4 } 6 \sin x d x$