Evaluate the definite integral
$\int _ {3}^{ 7 } \frac { 4 } { u ^ { 3 } } d u$. Use a graphing utility to verify your results.
A)$\frac { 40 } { 1,323 }$
B) $\frac { 160 } { 1,323 }$
C) $\frac { 20 } { 441 }$
D) $\frac { 80 } { 441 }$
E) $\frac { 160 } { 441 }$

Question

Evaluate the definite integral 
$\int _ {3}^{ 7 } \frac { 4 } { u ^ { 3 } } d u$. Use a graphing utility to verify your results. 
A)$\frac { 40 } { 1,323 }$
B) $\frac { 160 } { 1,323 }$
C) $\frac { 20 } { 441 }$
D) $\frac { 80 } { 441 }$
E) $\frac { 160 } { 441 }$

Quizplus · Accepted Answer

We can use the formula for integrating a reciprocal function, which is $\int \frac{1}{x^n} dx = \frac{-1}{(n-1)x^{n-1}} + C$. Using this formula with $n=3$, we get $\int \frac{1}{u^3} du = \frac{-1}{2u^2} + C$. Evaluating this between 3 and 7 gives us $\frac{-1}{2(7)^2} - \frac{-1}{2(3)^2} = \frac{80}{441}$.

Using a graphing utility, we can confirm this result by graphing $y=\frac{4}{x^3}$ and finding the area under the curve between $x=3$ and $x=7$ using the appropriate function on the calculator. This will give us a decimal approximation, which we can compare to the answer choices. The closest choice is D) $\frac{80}{441}$.

Evaluate the Definite Integral ∫374u3du\int _ {3}^{ 7 } \frac { 4 } { u ^ { 3 } } d u∫37​u34​du

Evaluate the Definite Integral $\int _ {3}^{ 7 } \frac { 4 } { u ^ { 3 } } d u$