Calculus Examples

$y = 7 x^{2}$ , $x = 1$ , $x = 4$

Step 1

Solve by substitution to find the intersection between the curves.

Tap for more steps...

Step 1.1

Eliminate the equal sides of each equation and combine.

$7 x^{2} = 0$

Step 1.2

Solve $7 x^{2} = 0$ for $x$ .

Tap for more steps...

Step 1.2.1

Divide each term in $7 x^{2} = 0$ by $7$ and simplify.

Tap for more steps...

Step 1.2.1.1

Divide each term in $7 x^{2} = 0$ by $7$ .

$\frac{7 x^{2}}{7} = \frac{0}{7}$

Step 1.2.1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 1.2.1.2.1.1

Cancel the common factor.

$\frac{7 x^{2}}{7} = \frac{0}{7}$

Step 1.2.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{7}$

Step 1.2.1.3

Simplify the right side.

Tap for more steps...

Step 1.2.1.3.1

Divide $0$ by $7$ .

$x^{2} = 0$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 1.2.3

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 1.2.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3

Substitute $0$ for $x$ .

$y = 0$

Step 1.4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 0)$

Step 2

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.

$A r e a = \int_{1}^{4} 7 x^{2} d x - \int_{1}^{4} 0 d x$

Step 3

Integrate to find the area between $1$ and $4$ .

Tap for more steps...

Step 3.1

Combine the integrals into a single integral.

$\int_{1}^{4} 7 x^{2} - (0) d x$

Step 3.2

Subtract $0$ from $7 x^{2}$ .

$\int_{1}^{4} 7 x^{2} d x$

Step 3.3

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int_{1}^{4} x^{2} d x$

Step 3.4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$7 (\frac{1}{3} x^{3}]_{1}^{4})$

Step 3.5

Simplify the answer.

Tap for more steps...

Step 3.5.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$7 (\frac{x^{3}}{3}]_{1}^{4})$

Step 3.5.2

Substitute and simplify.

Tap for more steps...

Step 3.5.2.1

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $1$ .

$7 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 3.5.2.2

Simplify.

Tap for more steps...

Step 3.5.2.2.1

Raise $4$ to the power of $3$ .

$7 (\frac{64}{3} - \frac{1^{3}}{3})$

Step 3.5.2.2.2

One to any power is one.

$7 (\frac{64}{3} - \frac{1}{3})$

Step 3.5.2.2.3

Combine the numerators over the common denominator.

$7 \frac{64 - 1}{3}$

Step 3.5.2.2.4

Subtract $1$ from $64$ .

$7 (\frac{63}{3})$

Step 3.5.2.2.5

Cancel the common factor of $63$ and $3$ .

Tap for more steps...

Step 3.5.2.2.5.1

Factor $3$ out of $63$ .

$7 \frac{3 \cdot 21}{3}$

Step 3.5.2.2.5.2

Cancel the common factors.

Tap for more steps...

Step 3.5.2.2.5.2.1

Factor $3$ out of $3$ .

$7 \frac{3 \cdot 21}{3 (1)}$

Step 3.5.2.2.5.2.2

Cancel the common factor.

$7 \frac{3 \cdot 21}{3 \cdot 1}$

Step 3.5.2.2.5.2.3

Rewrite the expression.

$7 (\frac{21}{1})$

Step 3.5.2.2.5.2.4

Divide $21$ by $1$ .

$7 \cdot 21$

Step 3.5.2.2.6

Multiply $7$ by $21$ .

$147$

Step 4