Question

The slope of the tangent to the curve y = f(x) at a point (x, y) is 2x + 1 and the curve passes through (1, 2). The area of the region bounded by the curves, the x-axis and the line x = 1 is

• A. 5/3 units
• B. 5/6 units
• C. 6/5 units
• D. 6 units

Answer 1

Answer: (B)

5/6 units

Answer 2

Since at point (x, y) of the curve, slope of the tangent to the curve is , so as given

= 2x + 1 Integrating, we get

y = x² + x + c But the curve passes through the point (1, 2), so we have

2 = 1 + 1 + c ̃ c = 0

\ equation of the curve is y = x² + x which is a parabola. Also the curve cuts x-axis at x = –1, and at x = 0. So the required area

= $\int _ { 0 } ^ { 1 } \left( x ^ { 2 } + x \right) d x$ = $\left( \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 2 } } { 2 } \right) _ { 0 } ^ { 1 }$ = $\frac { 1 } { 3 }$ + $\frac { 1 } { 2 }$