I have this question and It can simply be solved by using the formula of triangle, but the problem here we want to method of integration, is there is away to find the area of shaded area (Triangle) without using the formula of Triangle . any help appreciated.
[Math] Area under the line tangent to a curve at a point $(a, b)$
calculusintegration
Best Answer
We need the equation of the tangent line, whose slope is given by $m = f'(a)$, where $f'(a)$ is the derivative of the curve to which the line is tangent, evaluated at $x = a$.
Now, using the point $(a, b)$, the point of tangency, which happens to be a point on the line, to give us, along with slope, the equation of that line:
$$y - b = m(x-a) \iff y = mx + (b-ma)\iff y = m(x-a) + b$$
So we want the area under the line, evaluating the integral from $x = 0$ to the point where the line intersects the line $y = 0$, which is where $x = \dfrac{ma-b}{m} = a - \dfrac bm$
$$\int_0^{a - \large \frac bm} (m(x-a) +b)\,dx = \int_0^{a - \large\frac b{f'(a)}} \Big(f'(a)(x - a) + b\Big)\,dx$$