Here is a function,
to which we draw a tangent at x=3.
This way we get two regions.
One is bounded by the function, the tangent and the y axis,
the other one is bounded by the function, the tangent and the x axis.
We need to find the area of these regions.
Well, we will likely need the equation of the tangent.
Well, aren't we lucky? Here it comes:
Now let's get down to business.
It is much easier to calculate the area that is bounded by the y axis.
Since this is a normal region, it is sufficient to integrate the difference of the two functions:
Calculating the other area will be a lot more unpleasant.
First, we will need these intersections.
And now let's see the areas.
The area in question:
Area of a region bounded by a function and its tangent
Definite integration, Area under the curve of a function, Integrable, Primitive function, Newton-Leibniz formula, Fundamental Theorem of Calculus, Area of region that bounded two functions.