Problem 43 - Calculus and its applications

Question

Evaluate. $$ \int_{1}^{3}\left(3 t^{2}+7\right) d t $$

Step-by-Step Solution

Answer

The value of the definite integral is 40.

Step 1: Identify the Function to Integrate

The function given to us in the integral is \(3t^2 + 7\). This is a polynomial function, and we need to integrate it with respect to \(t\) from the lower limit of \(t=1\) to the upper limit of \(t=3\).

Step 2: Integrate the Polynomial Function

To find the indefinite integral of \(3t^2 + 7\), apply the power rule of integration. The power rule states that the integral of \(t^n\) is \(\frac{t^{n+1}}{n+1}\). Therefore, the function \(3t^2\) becomes \(\frac{3t^3}{3}\), which simplifies to \(t^3\), and the integral of the constant 7 is \(7t\). Thus, the indefinite integral is \(t^3 + 7t + C\), where \(C\) is the constant of integration.

Step 3: Evaluate the Indefinite Integral at the Bounds

The next step is to evaluate the definite integral \(\int_{1}^{3} (3t^2 + 7) \, dt\). First, substitute \(t=3\) into the indefinite integral to get \((3)^3 + 7(3)\), computed as \(27 + 21 = 48\). Then, substitute \(t=1\) to get \(1^3 + 7(1)\), computed as \(1 + 7 = 8\).

Step 4: Calculate the Definite Integral

Finally, subtract the value of the integral at the lower limit from the value at the upper limit: \(48 - 8\). This results in \(40\), which is the value of the definite integral.

Key Concepts

Polynomial Integration

Polynomial integration involves finding the integral of polynomial expressions. Polynomials consist of terms that are the sum of constants and variables raised to powers. For integration purposes, we're mostly concerned with these variable terms. It simplifies to adding together separate expressions for each part of the polynomial after integrating each term individually.

• Each term of the polynomial can be integrated separately.
• Always look for the sum of the separate integrals.
• Remember to add the constant of integration (C) for indefinite integrals.

Consider the polynomial function in the example, \(3t^2 + 7\). This consists of two terms: \(3t^2\) and \(7\). Integrating this involves dealing with each part: the squared term and the constant. When combined, these integrals together will give you the integrated polynomial with the constant term added for indefinite integrals. When integrating definite boundaries, you will eventually need to evaluate the resultant expressions at these limits and subtract.

Power Rule of Integration

The power rule is a straightforward method mainly used for integrating monomials, or expressions where the variable is raised to a power. It states that for any integer \( n \), \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \). This replaces the power of the term with one power higher, turning \( t^n \) into \( \frac{t^{n+1}}{n+1} \).

• Apply the power rule to each term in the polynomial.
• Do not forget to include the integration constant \( C \).
• Before using this rule, ensure \( n \) is not equal to \(-1\), as it requires a different approach (the logarithmic rule of integration).

In our integral \(3t^2+7\), the term \(3t^2\) becomes \(t^3\), since \(\frac{3t^3}{3}\) simplifies to \(t^3\). For the constant term \(7\), integration adds the variable \(t\) to give \(7t\). Each term's integration follows this power rule pattern, ensuring calculations are easy and efficient.

Evaluating Integrals

Once we establish the indefinite integral, evaluating it over specific limits involves plugging in values. In definite integrals, this means calculating the value of the integrated function at the upper limit and subtracting the function's value at the lower limit. This gives you the net "area" between the curve and the \(t\)-axis, between these two limits.
To evaluate \( \int_{1}^{3} (3t^2 + 7) \, dt \), you substitute the upper limit, \( t = 3 \), into the indefinite integral result, then subtract the result of substituting the lower limit, \( t = 1 \).

• Upper bound evaluation: Plug \( t = 3 \) into \(t^3 + 7t \).
• Lower bound evaluation: Substitute \( t = 1 \) into \(t^3 + 7t \).
• Subtract the results: \(48 - 8 = 40\).

By understanding this process, you can find definite integral values with confidence, as shown in our example where the result is \(40\). This value effectively represents the accumulated change between the specified bounds.