Which of the following statements are correct?
(A) \(\int_{0}^{1}\displaystyle x^{2}\,dx \lt \int_{0}^{1}\displaystyle x\,dx\)
(B) \(\int_{1}^{2}\displaystyle x^{2}\,dx \lt \int_{1}^{2}\displaystyle x\,dx\)
(C) \(\int_{0}^{\frac{\pi}{2}}\displaystyle\sin x\,dx = \int_{0}^{\frac{\pi}{2}}\displaystyle\cos x\,dx\)
(D) \(\int_{0}^{\pi}\displaystyle\sin x\,dx = 0\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
Working out each integral:
\( \displaystyle \int_{0}^{1}x^{2}\,dx = \frac{1}{3},\qquad \int_{0}^{1}x\,dx = \frac{1}{2} \)
\( \displaystyle \int_{1}^{2}x^{2}\,dx = \frac{8-1}{3} = \frac{7}{3},\qquad \int_{1}^{2}x\,dx = \frac{4-1}{2} = \frac{3}{2} \)
\( \displaystyle \int_{0}^{\pi/2}\sin x\,dx = \left[-\cos x\right]_{0}^{\pi/2} = 1,\qquad \int_{0}^{\pi/2}\cos x\,dx = \left[\sin x\right]_{0}^{\pi/2} = 1 \)
\( \displaystyle \int_{0}^{\pi}\sin x\,dx = \left[-\cos x\right]_{0}^{\pi} = 1 + 1 = 2 \)
So \(\frac{1}{3} \lt \frac{1}{2}\) on [0, 1], \(\frac{7}{3} \gt \frac{3}{2}\) on [1, 2], both trigonometric integrals on \([0, \frac{\pi}{2}]\) equal 1, and \(\int_{0}^{\pi}\sin x\,dx = 2\).
Checking the statements
(A) \(\int_{0}^{1}x^{2}\,dx \lt \int_{0}^{1}x\,dx\) — Correct. \(\frac{1}{3} \lt \frac{1}{2}\).
(B) \(\int_{1}^{2}x^{2}\,dx \lt \int_{1}^{2}x\,dx\) — Incorrect. \(\frac{7}{3} \gt \frac{3}{2}\), since \(x^{2} \gt x\) for \(x \gt 1\).
(C) \(\int_{0}^{\pi/2}\sin x\,dx = \int_{0}^{\pi/2}\cos x\,dx\) — Correct. Both equal 1.
(D) \(\int_{0}^{\pi}\sin x\,dx = 0\) — Incorrect. The value is \(1 + 1 = 2\).