Find the area of the region specified by the integral(s).
-$$\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { \ln x } d y d x$$
A) ln 6 + 1
B) 6(ln 6 - 1)
C) 6 ln 6
D) ln 6

Question

Quizplus · Accepted Answer

The integral represents the area of the region that lies under the curve $y=\ln x$ and above the x-axis in the first quadrant. 
To solve the integral, we can integrate with respect to $y$ first and then $x$:
$$\int_0^6\int_0^{\ln x} dydx$$

$$=\int_0^6 (\ln x - 0)dx$$

$$=[x\ln x -x]_0^6$$

$$=6\ln 6 - 6$$

Therefore, the area of the region is $6\ln 6 - 6$, which is equivalent to option B.

Find the Area of the Region Specified by the Integral(s) ∫06∫0ln⁡xdydx\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { \ln x } d y d x∫06​∫0lnx​dydx

Find the Area of the Region Specified by the Integral(s) $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { \ln x } d y d x$