Let R be the region bounded by the line y = 2x + 1, the x-axis and the lines x = 0 and x = 2. Which of the following statements are correct?
(A) The area of R is \(\displaystyle\int_{0}^{2}(2x+1)\,dx\)
(B) The area of R is 6 sq. units
(C) R is a trapezium whose parallel sides have lengths 1 and 5
(D) The line x = 1 divides R into two parts of equal area
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
For \(0 \le x \le 2\) the line y = 2x + 1 is above the x-axis, so the area of R is the integral of 2x + 1:
\( \displaystyle A = \int_{0}^{2}(2x+1)\,dx = \left[x^{2}+x\right]_{0}^{2} = 6 \)
The vertical sides of R are at x = 0, where the height is 1, and at x = 2, where the height is 5. So R is a trapezium with parallel sides 1 and 5.
The line x = 1 cuts R into two parts:
\( \displaystyle \int_{0}^{1}(2x+1)\,dx = \left[x^{2}+x\right]_{0}^{1} = 2 \)
\( \displaystyle \int_{1}^{2}(2x+1)\,dx = 6-2 = 4 \)
So the area of R is 6, its parallel sides are 1 and 5, and the line x = 1 cuts it into parts of area 2 and 4.
Checking the statements
(A) The area of R is \(\displaystyle\int_{0}^{2}(2x+1)\,dx\) — Correct. The line is above the x-axis on [0, 2].
(B) The area of R is 6 sq. units — Correct. \(\left[x^{2}+x\right]_{0}^{2} = 6\).
(C) R is a trapezium whose parallel sides have lengths 1 and 5 — Correct. The heights at x = 0 and x = 2 are 1 and 5.
(D) The line x = 1 divides R into two parts of equal area — Incorrect. The two parts have areas 2 and 4.