Question

Suppose that f is continuous, 5∫-2 f(x)dx=11 and 5∫-2 f(x)dx=14 Find the value of the integral 2∫5 f(x)dx

Answer 1

C. 3

Explanation:

One of the (many) properties of definite integrals is

`\mbox{\large \int\limits _(a)^(b)f(x)\,dx + \int\limits _(b)^(c)f(x)\,dx = \int\limits }_(a)^(c)f(x)\,dx`

Therefore

`\mbox{\large \int\limits _(-2)^(5)f(x)\,dx + \int\limits _(5)^(2)f(x)\,dx = \int\limits }_(-2)^(2)f(x)\,dx`

Given

`\mbox{\large \int\limits _(-2)^(5)f(x)\,dx = 11}}}\para`

and

`\mbox{\large \int\limits _(-2)^(2)f(x)\,dx = 14}}`

We get

`\mbox{\large 11+ \int\limits _(5)^(2)f(x)\,dx} = 14`

Subtracting 11 from both sides we get

`\mbox{\large \int\limits _(5)^(2)f(x)\,dx} = 14 - 11 = 3\\`

Answer: Choice C which is 3