I'm given a multiple choice question stating:
It is known that $\int _0^4\:f\left(x\right)dx=6$. The value of $\int _3^7\:f\left(x-3\right)+2dx$ is:
A) 8
B) 10
C) 14
D) 16
E) 18
Sorry if it is a really easy question, I'm currently doing Year 11 Calculus so this is my first year doing it. I've been trying this all day but can't figure it out. Thanks!
$\endgroup$ 22 Answers
$\begingroup$If $x$ goes from 3 to 7, $x-3$ goes from 0 to 4. Thus$$\int_3^7(f(x-3)+2)\,dx=\int_0^4(f(x)+2)\,dx$$Now we can split the integral:$$=\int_0^4f(x)\,dx+\int_0^42\,dx$$The left one is given to be 6 and the right one is easily evaluated:$$=6+2\cdot(4-0)=14$$
$\endgroup$ $\begingroup$If we use linearity of the integral we see that
$$\int_{3}^{7}h(x)+g(x)dx = \int_{3}^{7}h(x)dx+\int_{3}^{7}g(x)dx$$
In your case this means that
$$\int_{3}^{7}f(x-3)+2dx = \int_{3}^{7}f(x-3)dx+\int_{3}^{7}2dx$$
Do you know how we can do variable substitution to relate the integral
$$\int_{3}^{7}f(x-3)dx$$
to
$$\int_{0}^{4}f(x)dx$$
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