Definite integral

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This article/section deals with mathematical concepts appropriate for late high school or early college.

A definite integral is an integral with upper and lower limits.

A definite integral.

Definite Integrals

A definite integral is the area under the curve between two points on the function. In the picture below, the yellow area is "positive" and the blue area is "negative". The integral is evaluated by adding the positive area together and subtracting the negative area.

If the function f(x) is real rather than complex, then the definite integral is also known as a Riemann integral.

Solving Definite Integrals

Solving a definite integral usually has two main steps: integration and subtraction.

Sometimes approximations, such as the Riemann Integral or Simpson's rule are used. These approximations are used when:

• The exact answer is not needed, only a close approximation. (Common in Engineering)
• The rule for integration is very complex. (Such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int e^{x^2}\,dx} )
• The rule for integration is simply unknown. (Such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \zeta(x)\,dx} , the Zeta function)

Example 1

A graph representation of $f(x)$ in Example 1.

The thin blue line is f(x).

The yellow is what we add to get the value of the integral. It is the area.

The vertical green line, the lower bound, is at -3 because the lower bound of the definite integral is located at -3.

Likewise, the red line is the upper bound, which is at 5.

Click on the image to get a better view.

This is a very simple definite integral:

$\int _{-3}^{5}x^{2}dx$

Integration

Using indefinite integration, it can be shown that:

$\int x^{2}dx={1 \over 3}x^{3}=F(x)$

Note that $F(x)$ is the indefinite integral of $f(x)$.

Subtraction

Now, plug $5$ and $-3$ into the new expression and subtract, as shown by the Fundamental Theorem of Calculus.

$F(5)={1 \over 3}5^{3}={125 \over 3}$

$F(-3)={1 \over 3}(-3)^{3}={-27 \over 3}$

And subtract:

${125 \over 3}-{-27 \over 3}={125 \over 3}+{27 \over 3}$

$={152 \over 3}$

Example 2

This is a more complex definite integral that requires partial fractions to solve:

$\int _{4}^{12}{\frac {3x+11}{x^{2}-x-6}}dx$

Integration

See the Partial fractions in integration page for how to integrate the above expression (it is the example).
As shown on the page mentioned above:

$\int {\frac {3x+11}{x^{2}-x-6}}dx=4ln|x-3|-ln|x+2|+c$

Subtraction

This means that we can now subtract:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left [ 4ln|12-3|-ln|12+2|+c \right ] - \left [ 4ln|4-3|-ln|4+2|+c \right ]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left [ 4ln|9|-ln|14|+c \right ] - \left [ 4ln|1|-ln|6|+c \right ]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 4ln|9|-ln|14| - 4ln|1| + ln|6|}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 4ln|9|-ln|14| + ln|6|}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \approx 7.94160...}

Note the following:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ln(1) = 0}
• The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} on each side cancels out because we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c-c}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ln|6|} becomes positive because it was Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -(-ln|6|)}

See also

• Integral
• Indefinite integral