In which of the following cases the value of the integral is more than 10?
(A) \( \int_{0}^{2}\displaystyle|x|\,dx \)
(B) \( \int_{0}^{2}\displaystyle(x+3)^2\,dx \)
(C) \( \int_{0}^{2}\displaystyle e^x\,dx \)
(D) \( \int_{0}^{2}\displaystyle(2x+5)\,dx \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) and (D) only
Each integral follows from the second fundamental theorem. On [0, 2], |x| = x, so
\( \displaystyle \int_{0}^{2}|x|\,dx=\left[\frac{x^2}{2}\right]_{0}^{2} \)
\( \displaystyle =2 \)
\( \displaystyle \int_{0}^{2}(x+3)^2\,dx=\left[\frac{(x+3)^3}{3}\right]_{0}^{2} \)
\( \displaystyle =\frac{125-27}{3} \)
\( \displaystyle =\frac{98}{3}\approx 32.7 \)
\( \displaystyle \int_{0}^{2}e^x\,dx=e^2-1 \)
\( \displaystyle \approx 7.39-1=6.39 \)
\( \displaystyle \int_{0}^{2}(2x+5)\,dx=\left[x^2+5x\right]_{0}^{2} \)
\( \displaystyle =4+10=14 \)
So the four integrals are 2, 98/3 (about 32.7), e2 − 1 (about 6.39) and 14.
Checking the statements
(A) \( \int_{0}^{2}\displaystyle|x|\,dx \) — Incorrect. The value is 2, which is less than 10.
(B) \( \int_{0}^{2}\displaystyle(x+3)^2\,dx \) — Correct. The value is 98/3, about 32.7, which is more than 10.
(C) \( \int_{0}^{2}\displaystyle e^x\,dx \) — Incorrect. The value is e2 − 1, about 6.39, which is less than 10.
(D) \( \int_{0}^{2}\displaystyle(2x+5)\,dx \) — Correct. The value is 14, which is more than 10.