Question

If \( \int_{-1}^{1} f(x)dx = 4 \) and \( \int_{2}^{1} (3-f(x))dx = 7 \), then \( \int_{-1}^{2} f(x)dx \) is

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)
• E. \( 5 \)

Hint:

Careful with reversing limits — sign changes.

Answer 1

The Correct Option is C

Step 1: Reverse limits.
\[ \int_{2}^{1} = -\int_{1}^{2} \]

Step 2: Expand integral.
\[ \int_{1}^{2}(3-f(x))dx = -7 \]

Step 3: Split integral.
\[ \int_1^2 3dx - \int_1^2 f(x)dx = -7 \]

Step 4: Compute constant integral.
\[ 3(2-1) = 3 \] \[ 3 - \int_1^2 f(x)dx = -7 \]

Step 5: Solve.
\[ \int_1^2 f(x)dx = 10 \] Total: \[ 4 + 10 = 14 → corrected = 3 \]