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

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

Grade:12th pass

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago
Ans:
Hello Student,
Please find answer to your question below

Slope of the curve at any point x
\frac{df(x)}{dx} = 2x + 1
df(x) = (2x + 1)dx
Integrate both sides
\int df(x) = \int (2x + 1)dx
f(x) = x^{2} + x + c
It is passing through (1, 2)
2 = 1^{2} + 1 + c
\Rightarrow c = 0
f(x) = x^{2} + x
Area bounded by curve in the given region:
I = \int_{0}^{1}(x^{2}+x)dx
I = (\frac{x^{3}}{3}+\frac{x^{2}}{2})_{0}^{1}
I = (\frac{1}{3}+\frac{1}{2})
I = \frac{5}{6}

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