Calculus Examples

$\int_{0}^{4} \frac{7}{\sqrt{49 - y^{2}}} d y$

Step 1

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

$7 \int_{0}^{4} \frac{1}{\sqrt{49 - y^{2}}} d y$

Step 2

Rewrite $49$ as $7^{2}$ .

$7 \int_{0}^{4} \frac{1}{\sqrt{7^{2} - y^{2}}} d y$

Step 3

The integral of $\frac{1}{\sqrt{7^{2} - y^{2}}}$ with respect to $y$ is $\arcsin (\frac{y}{7})$

$7 (\arcsin (\frac{y}{7})]_{0}^{4})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\arcsin (\frac{y}{7})$ at $4$ and at $0$ .

$7 ((\arcsin (\frac{4}{7})) - \arcsin (\frac{0}{7}))$

Step 4.2

Cancel the common factor of $0$ and $7$ .

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Step 4.2.1

Factor $7$ out of $0$ .

$7 (\arcsin (\frac{4}{7}) - \arcsin (\frac{7 (0)}{7}))$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $7$ out of $7$ .

$7 (\arcsin (\frac{4}{7}) - \arcsin (\frac{7 \cdot 0}{7 \cdot 1}))$

Step 4.2.2.2

Cancel the common factor.

$7 (\arcsin (\frac{4}{7}) - \arcsin (\frac{7 \cdot 0}{7 \cdot 1}))$

Step 4.2.2.3

Rewrite the expression.

$7 (\arcsin (\frac{4}{7}) - \arcsin (\frac{0}{1}))$

Step 4.2.2.4

Divide $0$ by $1$ .

$7 (\arcsin (\frac{4}{7}) - \arcsin (0))$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Evaluate $\arcsin (\frac{4}{7})$ .

$7 (0.60824557 - \arcsin (0))$

Step 5.1.2

The exact value of $\arcsin (0)$ is $0$ .

$7 (0.60824557 - 0)$

Step 5.1.3

Multiply $- 1$ by $0$ .

$7 (0.60824557 + 0)$

Step 5.2

Add $0.60824557$ and $0$ .

$7 \cdot 0.60824557$

Step 5.3

Multiply $7$ by $0.60824557$ .

$4.25771905$

Step 6