What is the area of a region bounded by the curve $y=e^x$ and the lines $y=1$ and $x=1$?

Question

What is the area of a region bounded by the curve $y=e^x$ and the lines $y=1$ and $x=1$?

When $x=1$, $y=e$. When $y=1$, $x=0$.

I tried to find the area by saying that $A=\int^1_0 e^x dx= e - 1$.

However, the correct area is $e-2$. What am I doing wrong?

Answer 1

The correct integral is $$\int^1_0 (e^x-1) dx= e - 2$$

because the area is below the curve $e^x$ and above the line $y=1$. (Instead, you were calculating the area below the curve $e^x$ and above the $x$-axis.)

Answer 2

You omitted that fact that the region, let's call it $R$ is limited below by $y=1$.

So you need to substract $\int_0^1 1 \ dx = 1$ to your result.

Answer 3

Firstly draw the curves, then you will see that area is given by $$ \int_{x=0}^{x=1} \int_{y=1}^{\text{e}^x} \text{dy}\text{dx} =\text{e}-2$$