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

Step 1: Understanding the Concept:
This problem requires simplifying a definite integral by using a simple variable substitution to shift the limits of integration.

Step 2: Key Formula or Approach:
Use the substitution method. Let a new variable equal the argument of the function, i.e., let $t = x + 6$. Then find $dt$ and update the lower and upper limits of integration accordingly.

Step 3: Detailed Explanation:
The given integral is: \[ I = \int_{a-6}^{b-6} f(x + 6) dx \] Let's use a substitution to simplify the argument of the function $f$. Let $t = x + 6$. Differentiating both sides with respect to $x$: \[ dt = dx \] Now, we must adjust the limits of integration to match the new variable $t$: - When the lower limit $x = a - 6$, the new limit is $t = (a - 6) + 6 = a$. - When the upper limit $x = b - 6$, the new limit is $t = (b - 6) + 6 = b$. Substitute $t$, $dt$, and the new limits into the integral: \[ I = \int_{a}^{b} f(t) dt \] In a definite integral, the variable of integration is a "dummy variable." We can freely change the symbol from $t$ back to $x$ without altering the value of the integral: \[ I = \int_{a}^{b} f(x) dx \]

Step 4: Final Answer:
The integral is equal to $\int_{a}^{b} f(x) dx$.