Question

find the area bounded by a parabola and an oblique line ​

Answer 1

Answer:

the area bounded by a parabola and an oblique line

Answer 2

Answer: Area bounded by a parabola and line is  $\frac{m^2}{2}+\frac{m^3}{3}$

Step-by-step explanation:

Step 1: The equation of parabola and line

Equation of parabola = $y = x^2$ //Parabola from the origin

Equation of the line  = y = mx // Line from the origin

Considering the line originating from origin and then it will intercept parabola as shown in figure

Step 2: Find the Limits

At the interception of line and parabola

y of parabola = y of line

$x^2 = mx\\x^2 - mx = 0\\x(x-m)=0\\x=0 \\x = m$

Therefore limits of x are from 0  to 1

We need to find the area under this limit

Step 3: Find the area

The area will be

$A = \int_0^m(x - x^2dx)\\A = [\frac{x^2}{2}]_0^m+[\frac{x^3}{3}]_0^m\\A = \frac{m^2}{2}+\frac{m^3}{3}$

The area bounded by a parabola and an oblique line $\frac{m^2}{2}+\frac{m^3}{3}$  ​

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