Calculus Examples

$\int_{0}^{- x^{2} - y^{2}} x d z$

Step 1

Apply the constant rule.

$x z]_{0}^{- x^{2} - y^{2}}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $x z$ at $- x^{2} - y^{2}$ and at $0$ .

$(x (- x^{2} - y^{2})) - x \cdot 0$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $0$ by $- 1$ .

$x (- x^{2} - y^{2}) + 0 x$

Step 2.2.2

Multiply $0$ by $x$ .

$x (- x^{2} - y^{2}) + 0$

Step 2.2.3

Add $x (- x^{2} - y^{2})$ and $0$ .

$x (- x^{2} - y^{2})$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$x (- x^{2}) + x (- y^{2})$

Step 3.2

Rewrite using the commutative property of multiplication.

$- x \cdot x^{2} + x (- y^{2})$

Step 3.3

Rewrite using the commutative property of multiplication.

$- x \cdot x^{2} - x y^{2}$

Step 3.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.4.1

Move $x^{2}$ .

$- (x^{2} x) - x y^{2}$

Step 3.4.2

Multiply $x^{2}$ by $x$ .

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Step 3.4.2.1

Raise $x$ to the power of $1$ .

$- (x^{2} x^{1}) - x y^{2}$

Step 3.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{2 + 1} - x y^{2}$

Step 3.4.3

Add $2$ and $1$ .

$- x^{3} - x y^{2}$