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Calculate Integral of 4x X-A X-B for A B

Reviewed by Calculator Editorial Team

This guide explains how to calculate the definite integral of 4x(x-a)(x-b) from a to b. We'll cover the step-by-step process, provide a working calculator, and explain how to interpret the results.

How to Calculate the Integral

Calculating the integral of 4x(x-a)(x-b) from a to b involves several algebraic steps before applying integration rules. Here's the step-by-step process:

  1. First, expand the expression 4x(x-a)(x-b).
  2. Multiply the terms to get a cubic polynomial.
  3. Integrate the resulting polynomial term by term.
  4. Evaluate the antiderivative at the bounds a and b.
  5. Subtract the lower bound evaluation from the upper bound evaluation.

The final result will be a numerical value representing the area under the curve of 4x(x-a)(x-b) between x = a and x = b.

Formula

The integral of 4x(x-a)(x-b) from a to b can be calculated using the following steps:

  1. Expand 4x(x-a)(x-b) to get 4x³ - 4(a+b)x² + 4abx.
  2. Integrate term by term to get the antiderivative: x⁴ - (a+b)x³ + abx².
  3. Evaluate the antiderivative at b and a.
  4. Subtract the evaluation at a from the evaluation at b.

The final formula is:

[b⁴ - (a+b)b³ + ab²²] - [a⁴ - (a+b)a³ + aba²]

Worked Example

Let's calculate the integral of 4x(x-2)(x-3) from 2 to 3.

  1. First, expand 4x(x-2)(x-3):
  2. 4x(x² - 5x + 6) = 4x³ - 20x² + 24x
  3. Integrate term by term:
  4. x⁴ - (20/3)x³ + 12x² + C
  5. Evaluate at 3 and 2:
  6. At x=3: 81 - (20/3)(27) + 12(9) = 81 - 180 + 108 = 9
  7. At x=2: 16 - (20/3)(8) + 12(4) = 16 - 53.33 + 48 ≈ 10.67
  8. Subtract: 9 - 10.67 ≈ -1.67

The integral of 4x(x-2)(x-3) from 2 to 3 is approximately -1.67.

Interpreting the Result

The result of the integral represents the net area under the curve of 4x(x-a)(x-b) between x = a and x = b. A positive result indicates more area above the x-axis, while a negative result indicates more area below the x-axis.

In the example above, the negative result suggests that the curve dips below the x-axis more than it rises above it between x=2 and x=3.

FAQ

What does the integral represent?

The integral represents the net area under the curve of 4x(x-a)(x-b) between x = a and x = b. It can be positive or negative depending on where the curve lies relative to the x-axis.

Can I calculate this integral without expanding first?

While you can use integration by parts, expanding the expression first is generally simpler for this type of problem.

What if a and b are the same?

The integral will be zero because the upper and lower bounds are the same.