Calculus Examples

$f (x) = 2 x + 3$ , $[2, 8]$ , $n = 3$

Step 1

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{2}^{8} 2 x + 3 d x - \int_{2}^{8} 3 d x$

Step 2

Integrate to find the area between $2$ and $8$ .

Tap for more steps...

Step 2.1

Combine the integrals into a single integral.

$\int_{2}^{8} 2 x + 3 - (3) d x$

Step 2.2

Multiply $- 1$ by $3$ .

$\int_{2}^{8} 2 x + 3 - 3 d x$

Step 2.3

Combine the opposite terms in $2 x + 3 - 3$ .

Tap for more steps...

Step 2.3.1

Subtract $3$ from $3$ .

$2 x + 0$

Step 2.3.2

Add $2 x$ and $0$ .

$2 x$

$\int_{2}^{8} 2 x d x$

Step 2.4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{2}^{8} x d x$

Step 2.5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$2 (\frac{1}{2} x^{2}]_{2}^{8})$

Step 2.6

Simplify the answer.

Tap for more steps...

Step 2.6.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$2 (\frac{x^{2}}{2}]_{2}^{8})$

Step 2.6.2

Substitute and simplify.

Tap for more steps...

Step 2.6.2.1

Evaluate $\frac{x^{2}}{2}$ at $8$ and at $2$ .

$2 ((\frac{8^{2}}{2}) - \frac{2^{2}}{2})$

Step 2.6.2.2

Simplify.

Tap for more steps...

Step 2.6.2.2.1

Raise $8$ to the power of $2$ .

$2 (\frac{64}{2} - \frac{2^{2}}{2})$

Step 2.6.2.2.2

Cancel the common factor of $64$ and $2$ .

Tap for more steps...

Step 2.6.2.2.2.1

Factor $2$ out of $64$ .

$2 (\frac{2 \cdot 32}{2} - \frac{2^{2}}{2})$

Step 2.6.2.2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.6.2.2.2.2.1

Factor $2$ out of $2$ .

$2 (\frac{2 \cdot 32}{2 (1)} - \frac{2^{2}}{2})$

Step 2.6.2.2.2.2.2

Cancel the common factor.

$2 (\frac{2 \cdot 32}{2 \cdot 1} - \frac{2^{2}}{2})$

Step 2.6.2.2.2.2.3

Rewrite the expression.

$2 (\frac{32}{1} - \frac{2^{2}}{2})$

Step 2.6.2.2.2.2.4

Divide $32$ by $1$ .

$2 (32 - \frac{2^{2}}{2})$

Step 2.6.2.2.3

Raise $2$ to the power of $2$ .

$2 (32 - \frac{4}{2})$

Step 2.6.2.2.4

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 2.6.2.2.4.1

Factor $2$ out of $4$ .

$2 (32 - \frac{2 \cdot 2}{2})$

Step 2.6.2.2.4.2

Cancel the common factors.

Tap for more steps...

Step 2.6.2.2.4.2.1

Factor $2$ out of $2$ .

$2 (32 - \frac{2 \cdot 2}{2 (1)})$

Step 2.6.2.2.4.2.2

Cancel the common factor.

$2 (32 - \frac{2 \cdot 2}{2 \cdot 1})$

Step 2.6.2.2.4.2.3

Rewrite the expression.

$2 (32 - \frac{2}{1})$

Step 2.6.2.2.4.2.4

Divide $2$ by $1$ .

$2 (32 - 1 \cdot 2)$

Step 2.6.2.2.5

Multiply $- 1$ by $2$ .

$2 (32 - 2)$

Step 2.6.2.2.6

Subtract $2$ from $32$ .

$2 \cdot 30$

Step 2.6.2.2.7

Multiply $2$ by $30$ .

$60$

Step 3