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Chapter 5: Problem 39

Evaluate. $$ \int_{a}^{b} 3 t^{2} d t $$

Short Answer

Expert verified
The integral evaluates to \[b^3 - a^3\].

Step by step solution

01

Understand the Integral

The exercise asks to evaluate the definite integral of the function \[- 3t^2 \] from \[t = a \] to \[t = b \]. This involves finding the antiderivative (indefinite integral) of the function, and then applying the limits of integration.
02

Find the Antiderivative

To find the antiderivative, use the power rule for integration. The integral of \[t^n \] is \[\frac{t^{n+1}}{n+1}\]. Thus, the antiderivative of \[3t^2\] is: \[\int 3t^2 \, dt = 3 \cdot \frac{t^{3}}{3} = t^3\].
03

Apply the Limits of Integration

Apply the Fundamental Theorem of Calculus, which states that: \[ \int_a^b f(t) \, dt = F(b) - F(a) \], where \(F(t)\) is the antiderivative of \(f(t)\). Here, \(F(t) = t^3\). So, \[\int_a^b 3t^2 \, dt = t^3 \bigg|_a^b = b^3 - a^3 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative
An antiderivative is a function whose derivative is the original function you started with. Finding the antiderivative is also known as integration. In our problem, we want to find the antiderivative of \(3t^2\). For each function, there can be many antiderivatives, differing by a constant. For example, the antiderivative of \(3t^2\) can be \(t^3 + C\), where \(C\) represents any constant.
The general form of an antiderivative is crucial because it helps in understanding how functions grow and change. It is fundamental in solving real-world problems involving areas under curves and in physics for concepts like velocity and acceleration.
Power Rule
The power rule for integration is a method used to find the antiderivative of functions of the form \(t^n\). This rule states that if \(n\) is any real number not equal to -1, then the integral of \(t^n\) with respect to \(t\) is:
\[ \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \] In our exercise, we apply this rule to \(3t^2\). This involves increasing the exponent by one and then dividing by this new exponent:
\[ \int 3t^2 \, dt = 3 \frac{t^{3}}{3} = t^3. \]
This is because the derivative of \(t^3\) gives us back \(3t^2\), confirming our antiderivative is correct.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus bridges the concept of differentiation and integration. It states that if \(F(t)\) is an antiderivative of \(f(t)\), then the definite integral of \(f(t)\) from \(a\) to \(b\) is given by:
\[ \int_a^b f(t) \, dt = F(b) - F(a). \] In our problem, \(F(t) = t^3\) is the antiderivative of \(3t^2\). To evaluate the definite integral:\[ \int_a^b 3t^2 \, dt = t^3 \bigg|_a^b = b^3 - a^3. \] This means we substitute the upper limit \(b\) into our antiderivative \(t^3\), and then subtract the result of substituting the lower limit \(a\). This process gives us the exact area under the curve \(3t^2\) from \(t = a\) to \(t = b\).

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