Areas with exponential functions

Finding areas under exponential curves and between exponential curves and other curves or lines.

Areas under the curve

$y = e^x$ is always above the $x$-axis, so its integral is always an area and no splitting is needed.

Example 1●

Find the area under $y = e^x$ from $x = 0$ to $x = 1$.

$\Big[e^x\Big]_0^1 = e - 1$ square units.

Example 2●●

Find the area bounded by $y = e^x$, the $y$-axis and the line $y = e$.

The curve meets $y = e$ at $x = 1$, and the line is on top.

$\displaystyle\int_0^1 (e - e^x)\,dx = \Big[ex - e^x\Big]_0^1 = 0 - (-1) = 1$ square unit.

Areas between curves

Between y = eˣ and y = e⁻ˣ

For x ≥ 0, y = eˣ is the upper curve.

Example 3●●

Find the area between $y = e^x$ and $y = e^{-x}$ from $x = 0$ to $x = 1$.

$\displaystyle\int_0^1 \big(e^x - e^{-x}\big)\,dx = \Big[e^x + e^{-x}\Big]_0^1 = e + \tfrac1e - 2$ square units.

Example 4●●●

The line $y = x + 1$ is the tangent to $y = e^x$ at $x = 0$. Find the area between them from $x = 0$ to $x = 1$.

The curve is concave up, so it lies above its tangent.

$\displaystyle\int_0^1 \big(e^x - x - 1\big)\,dx = \Big[e^x - \tfrac{x^2}{2} - x\Big]_0^1 = \left(e - \tfrac32\right) - 1 = e - \tfrac52$ square units.

Flashcards

Area under $y = e^x$ from $0$ to $1$

$e - 1$

Does $y = e^x$ ever need splitting?

No, it is always above the $x$-axis.

Area between curves

$\displaystyle\int (\text{upper} - \text{lower})\,dx$

Practice

Find the area under $y = e^x$ from $x = 0$ to $x = 1$. Type $e$ as e.
Find the area bounded by $y = e^x$, the $y$-axis and $y = e$.
Find the area between $y = e^x$ and $y = e^{-x}$ from $x = 0$ to $x = 1$.
Find the area between $y = e^x$ and $y = x + 1$ from $x = 0$ to $x = 1$.
Find the area under $y = e^{2x}$ from $x = 0$ to $x = \ln 3$.
Find the area under $y = e^{-x}$ from $x = 0$ to $x = \ln 2$.
Find the area between $y = e^x$ and $y = 1$ from $x = 0$ to $x = \ln 4$. Type $\ln 4$ as ln4.
The area under $y = e^x$ from $x = -1$ to $x = 1$ is:
Why does an integral of $e^{ax + b}$ never need splitting at the $x$-axis?