Question

$\int_{a-6}^{b-6} f(x + 6) dx$ is equal to

• A. $\int_{a}^{b} f(x - 6) dx$
• B. $\int_{a}^{b} f(x + 6) dx$
• C. $\int_{a}^{b} f(x) dx$
• D. $\int_{a}^{b} f(-x) dx$

Hint:

This demonstrates the translation property of definite integrals: Shifting the function horizontally by $c$ ($f(x+c)$) and shifting the integration limits by the opposite amount ($-c$) results in the same area under the curve. $\int_{A-c}^{B-c} f(x+c)dx = \int_{A}^{B} f(x)dx$.

Answer 1

The Correct Option is C

The problem involves evaluating the integration of a function with a shifted argument and verifying its equivalency with a known integral limit transformation.

We are given:

\(\int_{a-6}^{b-6} f(x + 6) \, dx\).

We need to show that this integral is equal to \(\int_{a}^{b} f(x) \, dx\).

Change of Variables in Integration:

Let us use a substitution to simplify this integral:

1. Set u = x + 6. Then, du = dx.
2. When x = a - 6, then u = (a - 6) + 6 = a.
3. When x = b - 6, then u = (b - 6) + 6 = b.

Substituting these values into the integral:

\(\int_{a-6}^{b-6} f(x + 6) \, dx = \int_{a}^{b} f(u) \, du\).

Therefore, this integral transforms to:

\(\int_{a}^{b} f(x) \, dx\), where we've replaced the dummy variable u with x.

Conclusion:

Thus, the correct option is:

\(\int_{a}^{b} f(x) \, dx\).

Therefore, the answer to the problem is \(\int_{a}^{b} f(x) \, dx\).