If we know that the f is continuous and integral 0 to 4&nbsp;f(x)&nbsp;dx&nbsp;=10, compute the integral 0 to 2&nbsp;f(2x)&nbsp;dx&nbsp;

We will start working with the integral ∫02f(2x)dx Substitute 2x=u and 2dx=du =12∫02f(2x)(2dx) Limits of integration will change from ∫02 to ∫2×02×2=∫04 12∫04f(u)du Remember that ∫abf(u)du=∫abf(x)dx =12∫04f(x)dx It is given that ∫04f(x)dx=10 =12×10=5 Result: ∫02f(2x)dx=5

Question: "If we know that the f is continuous..."

College
Calculus and Analysis
Integral Calculus
Integrals

Cabiolab

Answered question

2021-10-07

If we know that the f is continuous and integral 0 to 4 $f (x) d x = 10$ , compute the integral 0 to 2 $f (2 x) d x$

Answer & Explanation

StrycharzT

Skilled2021-10-08Added 102 answers

We will start working with the integral

\int_{0}^{2} f (2 x) d x

Substitute

2 x = u

and

2 d x = d u

= \frac{1}{2} \int_{0}^{2} f (2 x) (2 d x)

Limits of integration will change from

\int_{0}^{2}

\int_{2 \times 0}^{2 \times 2} = \int_{0}^{4}

\frac{1}{2} \int_{0}^{4} f (u) d u

Remember that

\int_{a}^{b} f (u) d u = \int_{a}^{b} f (x) d x

= \frac{1}{2} \int_{0}^{4} f (x) d x

It is given that

\int_{0}^{4} f (x) d x = 10

= \frac{1}{2} \times 10 = 5

Result:

\int_{0}^{2} f (2 x) d x = 5

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