2t T 1 3 Integration Calculator
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Reviewed by Calculator Editorial Team
This calculator computes the definite integral of the function 2t³ + t - 3. You can specify the limits of integration and get both the exact result and a graphical representation of the function and its integral.
How to Use This Calculator
To calculate the integral of 2t³ + t - 3 between two limits:
- Enter the lower limit (a) in the first input field
- Enter the upper limit (b) in the second input field
- Click the "Calculate" button
- View the result and chart showing the function and its integral
The calculator will display the exact value of the integral and show a visual representation of the function and its antiderivative.
The Integration Formula
The integral of 2t³ + t - 3 from a to b is calculated using the antiderivative of the function:
∫(2t³ + t - 3) dt = (1/2)t⁴ + (1/2)t² - 3t + C
To find the definite integral from a to b, we evaluate the antiderivative at the upper and lower limits and subtract:
∫[a to b] (2t³ + t - 3) dt = [(1/2)b⁴ + (1/2)b² - 3b] - [(1/2)a⁴ + (1/2)a² - 3a]
This formula gives the exact area under the curve of the function between the specified limits.
Worked Example
Let's calculate the integral from t = 0 to t = 2:
∫[0 to 2] (2t³ + t - 3) dt = [(1/2)(2)⁴ + (1/2)(2)² - 3(2)] - [(1/2)(0)⁴ + (1/2)(0)² - 3(0)]
= [(1/2)(16) + (1/2)(4) - 6] - [0 + 0 - 0]
= [8 + 2 - 6] - 0 = 4
The integral from 0 to 2 is exactly 4. The chart in the calculator will show the function and its integral between these limits.
Practical Applications
Integrals of polynomials like 2t³ + t - 3 are used in various fields:
- Physics: Calculating work done by variable forces
- Engineering: Finding areas under curves in design
- Economics: Calculating total cost or revenue functions
- Mathematics: Understanding the behavior of functions
This calculator helps professionals and students quickly evaluate these integrals for their specific applications.
Frequently Asked Questions
What is the antiderivative of 2t³ + t - 3?
The antiderivative is (1/2)t⁴ + (1/2)t² - 3t + C, where C is the constant of integration.
Can I calculate the integral from negative to positive limits?
Yes, the calculator accepts any real number for the limits of integration.
What if the upper limit is less than the lower limit?
The calculator will return the negative of the integral from b to a.
Is the result always exact or can it be an approximation?
The result is always exact for polynomial functions like 2t³ + t - 3.