Areas Between Curves Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Mateo Johnson (Solvelet student)

Find the area between the curves

y = x^2

and

y = 2x + 3

from

x = -1

x = 3

Solution

Step 1: Identify the curves and the limits of integration. The curves are

y = x^2

and

y = 2x + 3

and the limits are

x = -1

x = 3

. Step 2: Set up the integral for the area between the curves:

\int_{-1}^{3} \left( (2x + 3) - x^2 \right) \, dx.

Step 3: Simplify the integrand:

\int_{-1}^{3} (2x + 3 - x^2) \, dx.

Step 4: Evaluate the integral:

\int_{-1}^{3} (2x + 3 - x^2) \, dx = \left[ x^2 + 3x - \frac{x^3}{3} \right]_{-1}^{3}.

Step 5: Apply the limits of integration:

\left[ 3^2 + 3(3) - \frac{3^3}{3} \right] - \left[ (-1)^2 + 3(-1) - \frac{(-1)^3}{3} \right] = \left[ 9 + 9 - 9 \right] - \left[ 1 - 3 + \frac{1}{3} \right].

= 9 - (-2 + \frac{1}{3}) = 9 + 2 - \frac{1}{3} = 11 - \frac{1}{3} = \frac{33}{3} - \frac{1}{3} = \frac{32}{3}.

Step 6: Conclusion. The area between the curves

y = x^2

and

y = 2x + 3

from

x = -1

x = 3

\frac{32}{3}

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Aiden Flores on Solvelet

1. Find the enclosed area by the curves

y = x^2

and

y = 2x - 1

2. Find the area of the bounded curves of regions

y = \sin(x)

and

y = \cos(x)

over the interval

[0, \pi]

DefinitionAn area between the curve is the region which is bound by two or more curves in a plane. This is determined by taking the definite integral of the upper curve minus lower curve over some range of values. This is a critical idea in calculus for finding the area under curves, the volumes of revolution, optimization problems. It can be found in many places in physics, engineering and economics, but is used in particular to measure regions surrounded by functions. Area between curves example: To calculate the area between two curves y=f(x) and y=g(x) over an interval [a,b], integral ( \int_a^b)

Related Calculators

Numerical Methods Calculator Geometric Sequences Calculator Complex Arithmetic Calculator Rational Expressions Calculator

Need topic explanation ? Get video explanation