Does $\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x$ express the area between the curve y = $e ^ { x }$ - 1 and the x-axis between x = 1 and $x = 3 ?$ Give a reason for your answer.
A) Yes, because y = $e ^ { x }$ - 1 does not cross the x-axis between 1 and 3 .
B) Yes, because y = $e ^ { x }$ - 1 is nonnegative for all x between 1 and 3 .
C) No, because $\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x$ is negative.
D) No, because y = $e ^ { x }$ - 1 crosses the x-axis between 1 and 3 .
E) none of these

Question

Quizplus · Accepted Answer

The function $y = e^x - 1$ is nonnegative for all $x > 0$, including the interval from 1 to 3, because $e^x > 1$ for any $x > 0$. Therefore, the integral represents the area between the curve and the x-axis in this interval.

Does ∫13(ex−1)dx\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x∫13​(ex−1)dx

Does $\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x$