**Math Calculators** ▶ Area Under The Curve Calculator

An online area under the curve calculator provides the area for the given curve function specified with the upper and lower limits. This area under curve calculator displays the integration with steps and integrates the function term-by-term. So, read on to learn how to find the area under the curve by integration, how to approximate area under curve using its formula, and much more.

In mathematics, the area under the curve for the given function f(x) having the limits x = a and x = b is given by the definite integral formula. In case, if the area between two bounding values lies above the x-axis, then it has a positive sign. If the area between two values lies below the x-axis, then the negative sign has to be taken.

The area under a curve between two points is found out by doing a definite integral between the two points. To find area under curve y = f(x) between x = a & x = b, you need to integrate y = f(x) between the limits of a and b. This area can be calculated using integration with given limits.

However, use our online Area Between Two Curves Calculator to find the area between two curves on a given interval corresponding to the difference between the definite integrals.

The formula for AUC = \( ∫^a_b f (x) dx \)

Where,

A and b are upper and lower limits,

F(x) is curve function.

**Example:**

Compute the AUC of the function, f(x) = 6x + 3, the limit is given as x = 0 to 4.

**Solution:**

Calculating area under curve for given function: f(x) = 6x + 3

Upper Limit: 4

Lower Limit: 0

Now, the area under the curve calculator substitute the curve function in the equation:

$$ ∫^4_0 (6x + 3) dx $$

Then, the area under parametric curve calculator integrates the function term-by-term:

First, take the integral of a function:

$$ ∫ 6x dx = 6 ∫ x dx $$

The integral of \( x^n is x^{n+1} / n + 1 \), when n ≠−1:

$$ ∫ x dx = x^2 / 2 $$

So, the result is: \( 3 x^2 \)

Then, area under a curve calculator take the integral of a constant:

$$ ∫ 3 dx = 3x $$

The result is: \( 3x^2 + 3x \)

**Simplify: **

$$ 3x ( x + 1) $$

Add the constant of integration:

$$ 3x (x + 1) + constant $$

Now, estimate AUC by substitute the upper and lower limit values in the obtained equation:

Add x = 4 and 2, So:

$$ 3(4) ( 4 + 1) – 3(2) (2 + 1) $$

$$ 12 ( 5) – 6 (3) = 60 – 18 $$

Thus, AUC is 42.

Additionally, an Online Integral Calculator helps you to evaluate the integrals of the functions with respect to the variable involved.

The area under a given function has an upper limit and a lower limit, which are determined by the integrations. In order to find area under the curve by hand, you should stick to the following step-by-step guidelines:

- Take any function f(x) and limit x = m, x = n
- Perform integration on the function with upper limit n and lower limit m.
- Calculate the points and enter the values a and b.
- Subtract f(n) from f(m) to obtain the results.

However, if you want to do risk-free calculations, then area under the curve calculator find the area in a fraction of a second.

**Example**

Calculate area under curve \( y = x^3 + 5 \) with limits x = 0 to x = 1?

**Solution:**

Given function is: \( y = x^3 + 5 \)

$$ Area = ∫_0^1 x^3 + 5 dx $$

$$ = x (x^3 + 20) / 4 $$

$$ = 1 ((1)^3 + 20) / 4 – 0 ((0)^3 + 0) / 4 $$

$$ = 21 + 0 / 4 $$

$$ Area = 21/4 $$

An online area under a curve calculator finds the area under curve with integration by following these guidelines:

- Firstly, enter a curve function.
- Now, substitute the upper and lower limit.
- Then, select the variable for integration from the drop-down list.
- Hit the calculate button to compute the AUC.

- The area under curve calculator provides the area and integral steps.
- In the integral steps, the calculator integrates the curve function term-by-term for obtaining the results.

The area cannot be negative. If the problem is to find the value of the integral, the result may be negative.

Find the area between two points:

- We convert each original point number to a Z-point score.
- Find the range of two z values.
- Subtract the smallest area from the largest area.

The area under the normal distribution curve expresses probability and the total AUC sum. Most of the data values in a normal distribution tend to merge, and further a value from the mean.

The AUC is the magnitude of the displacement that is equal to the distance traveled, only for constant acceleration.

Use this online area under the curve calculator, which uses stepwise integration for computing the area under curve by given curve function. This calculator finds the area under the graph instantly using upper and lower limits.

From the source of Wikipedia: Area under the curve (pharmacokinetics), Interpretation and usefulness of AUC values, AUC and bioavailability, Absolute bioavailability.

From the source of Interactive Mathematics: UAC by Integration, Curves which are entirely above the x-axis, Curve from First Principles, Curves which are entirely below the x-axis.

From the source of Medium: Area using summation notation, Area using anti-derivatives, Area using numerical calculations, constant velocity.