Question

Evaluate the definite integral. $$ \int_{-1}^{1}(\sqrt[3]{t}-2) d t $$

Short answer

The result of the definite integral is \( -\frac{3}{2} \)

Step-by-step solution

Splitting the Integral

The integral can be separated into two simpler integrals. This is due to the additive property of integrals, which states that the integral of a sum of two functions is equal to the sum of the integrals of each function independently. Therefore, \[ \int_{-1}^{1} (\sqrt[3]{t}-2) dt = \int_{-1}^{1} \sqrt[3]{t} dt - \int_{-1}^{1} 2 dt\]

Integrating the Separated Functions

Separately integrating both functions: the integral of \[ \sqrt[3]{t} \] can be written as \[ t^{1/3} \] and its antiderivative is \( \frac{3}{4}t^{4/3} \) when evaluated from -1 to 1. For the integral of \[ 2 \] the antiderivative is \[ 2t \] when evaluated from -1 to 1.

Applying the Limits

When applying the limits of integration to the calculated antiderivatives, the result is:\[ \frac{3}{4}(1^{4/3}-(-1)^{4/3}) - 2(1-(-1)) \] Simplifying the terms we get that the final value equal to \( -\frac{3}{2} \)

Key concepts

Additive Property of Integrals

The additive property of integrals is a fundamental concept in calculus that states the integral of a sum of functions can be broken down into the sum of individual integrals. This property simplifies complex integrals and allows us to evaluate them piece by piece.

To understand this principle, imagine trying to find the total area under a composite curve. Instead of handling the entire curve all at once, you decompose it into simpler parts. Each simpler curve's area is found separately, and the results are added to get the total area.
For our original problem, the integral of the function \((\sqrt[3]{t} - 2)\) from -1 to 1 can be split into two simpler parts:

• \( \int_{-1}^{1} \sqrt[3]{t} \, dt \)
• \( \int_{-1}^{1} -2 \, dt \)

By applying the additive property, we handle each part separately, making the overall evaluation more straightforward.

Antiderivatives

Antiderivatives are the inverse operation of derivatives. Finding an antiderivative involves determining the original function before it was differentiated. This step is crucial in the process of evaluating definite integrals.

For the integral \( \int \sqrt[3]{t} \, dt \), the antiderivative is found by identifying the function whose derivative is \( \sqrt[3]{t} \). In this case, the antiderivative is \( \frac{3}{4}t^{4/3} \).
Similarly, for the integral \( \int 2 \, dt \), the antiderivative is \( 2t \).

When we compute definite integrals, antiderivatives are evaluated at the upper and lower limits of integration, and their difference gives the area under the curve between these limits. Antiderivatives simplify the calculation and are essential in solving integration problems.

Limits of Integration

Limits of integration in a definite integral are the boundaries between which we want to calculate the area under a curve. For our problem, the limits are -1 and 1.

The limits apply to the antiderivatives we have found. First, we substitute the upper limit into the antiderivative function, then subtract the result of substituting the lower limit.
For

• \( \frac{3}{4}((1)^{4/3} - (-1)^{4/3}) \) results in putting the limits into \( \frac{3}{4}t^{4/3} \)
• \( 2(1 - (-1)) \) for the constant part involves substituting limits into \( 2t \)

Evaluating these expressions results in the simplified value of the definite integral. Understanding and correctly applying the limits of integration is key to finding an accurate solution to the integral.