Use the following property of the definite integral to estimate the definite integral $\int _ { x / 6 } ^ { s / 4 } 6 \sin x d x$ : 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)
$\frac { \pi } { 2 } \leq \int _ { x / 6 } ^ { x / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 2 }$
B) $\frac { \sqrt { 2 } \pi } { 4 } \leq \int _ { x / 6 } ^ { z / 4 } 6 \sin x d x \leq \frac { \pi } { 4 }$
C)
$\frac { \pi } { 4 } \leq \int _ { x / 6 } ^ { \pi / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 4 }$
D) $\frac { \sqrt { 2 } \pi } { 2 } \leq \int _ { z / 6 } ^ { z / 4 } 6 \sin x d x \leq \frac { \pi } { 2 }$

Question

Use the following property of the definite integral to estimate the definite integral $\int _ { x / 6 } ^ { s / 4 } 6 \sin x d x$ : 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)
$\frac { \pi } { 2 } \leq \int _ { x / 6 } ^ { x / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 2 }$
B) $\frac { \sqrt { 2 } \pi } { 4 } \leq \int _ { x / 6 } ^ { z / 4 } 6 \sin x d x \leq \frac { \pi } { 4 }$
C)
$\frac { \pi } { 4 } \leq \int _ { x / 6 } ^ { \pi / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 4 }$
D) $\frac { \sqrt { 2 } \pi } { 2 } \leq \int _ { z / 6 } ^ { z / 4 } 6 \sin x d x \leq \frac { \pi } { 2 }$

Quizplus · Accepted Answer

The correct integral limits should be numbers, not variables, for a specific estimate. Option C correctly uses $\pi/4$ as the upper limit, which matches the format required for estimation using the given property. The sine function's values range between -1 and 1, but since we're looking at a positive interval (0 to $\pi/4$), the sine values will be between 0 and $\sqrt{2}/2$. Multiplying by 6 adjusts the range to 0 to $3\sqrt{2}$, and the interval length is $\pi/4 - \pi/6 = \pi/12$. Thus, the estimates should be based on these adjusted values and the correct interval.

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

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