Evaluate the integral by changing the order of integration in an appropriate way.
-$$\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { 35 } \int _ { y / 7 } ^ { 5 } \frac { \tan x } { x z } d x d y d z$$
A) - 7 ln 2 ln cos 5
B) - 7 ln 4 ln cos 5
C) 7 ln 4 ln cos 5
D) 7 ln 2 ln cos 5

Question

Quizplus · Accepted Answer

The integral given is $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { 35 } \int _ { y / 7 } ^ { 5 } \frac { \tan x } { x z } d x d y d z$. To evaluate this integral by changing the order of integration, we first need to understand the limits of integration for $x$, $y$, and $z$. The limits for $x$ are from $y/7$ to 5, for $y$ are from 0 to 35, and for $z$ are from 1 to 2.Changing the order of integration to $dx dy dz$ requires us to reconsider the bounds in terms of $x$ first. The $x$ bounds are already given as from $y/7$ to 5, which means $x$ ranges from a minimum of 0 (when $y=0$) to a maximum of 5. For $y$, given $x$ ranges from $y/7$ to 5, $y$ can be reinterpreted to range from 0 to $7x$, with the upper limit of $y$ being 35 when $x=5$. For $z$, the bounds remain unchanged as from 1 to 2.Thus, the integral becomes $\int_{0}^{5} \int_{0}^{7x} \int_{1}^{2} \frac{\tan x}{xz} dz dy dx$ when reordered. Evaluating the integral over $z$ first gives $\int_{0}^{5} \int_{0}^{7x} \frac{\tan x}{x} \ln(z) \Big|_{1}^{2} dy dx$, which simplifies to $\int_{0}^{5} \int_{0}^{7x} \frac{\tan x}{x} \ln(2) dy dx$. Integrating over $y$ next gives $\int_{0}^{5} \frac{\tan x}{x} \ln(2) \cdot 7x dx$, which simplifies further to $7 \ln(2) \int_{0}^{5} \tan x dx$.The integral of $\tan x$ from 0 to 5 is $\ln | \cos x | \Big|_{0}^{5}$, which gives $-\ln(\cos(5))$. Therefore, the final answer is $7 \ln(2) (-\ln(\cos(5))) = -7 \ln(2) \ln(\cos(5))$, which matches choice A.

Evaluate the Integral by Changing the Order of Integration in an Appropriate