Determine the area of ​​the shape bounded by the curves

Question

$$y=x^2 + 1, x=2, x=1, y=0$$

I've got exam today and I'm learning how to solve this type of task. The exam is about derivations mostly.

Answer 1

If you draw the curves roughly on a graph, you'll see that the required area is just the area under the curve $y=x^2+1$ between certain limits. Integrate $y=x^2+1$ between those limits.

Answer 2

Well in this case we want to integrate instead of taking the derivative.

Drawing the picture we get

To find the area under the curve, all we have to do is integrate - so we get

$$A=\int\limits_1^2(x^2+1)-0\,dx=\int\limits_1^2x^2+1\,dx=\frac{1}{3}x^3+x\bigg|_{x=1}^{x=2}=\frac{8}{3}+2-\frac{1}{3}-1=\frac{10}{3}$$

Answer 3

$$ \int_1^2 x^2+1 \mathrm{d}x = \left[\frac{x^3}{3}+x\right]_1^2=\frac{10}{3} $$