Problem 47 - Calculus and its applications

Question

Evaluate. $$ \int_{-2}^{5}\left(2 x^{2}-3 x+7\right) d x $$

Step-by-Step Solution

Answer

The value of the definite integral is \( \frac{1171}{6} \).

Step 1: Identify the Function to Integrate

The given definite integral is \( \int_{-2}^{5}(2x^2 - 3x + 7) \, dx \). This involves integrating the quadratic polynomial function \( 2x^2 - 3x + 7 \).

Step 2: Compute the Antiderivative

To integrate the polynomial function, find the antiderivative. The antiderivative of \( 2x^2 \) is \( \frac{2}{3}x^3 \), the antiderivative of \( -3x \) is \( -\frac{3}{2}x^2 \), and for the constant \( 7 \), it is \( 7x \). Thus, the antiderivative is \( \frac{2}{3}x^3 - \frac{3}{2}x^2 + 7x \).

Step 3: Evaluate the Antiderivative at the Upper Limit

Substitute the upper limit of the integral, \( x = 5 \), into the antiderivative: \( \frac{2}{3}(5)^3 - \frac{3}{2}(5)^2 + 7(5) \). Simplify to obtain \( \frac{2}{3}(125) - \frac{3}{2}(25) + 35 \), which equals \( \frac{250}{3} - \frac{75}{2} + 35 \).

Step 4: Evaluate the Antiderivative at the Lower Limit

Substitute the lower limit of the integral, \( x = -2 \), into the antiderivative: \( \frac{2}{3}(-2)^3 - \frac{3}{2}(-2)^2 + 7(-2) \). Simplify to obtain \( \frac{2}{3}(-8) - \frac{3}{2}(4) - 14 \), which equals \( -\frac{16}{3} - 6 - 14 \).

Step 5: Compute the Definite Integral

Calculate the definite integral by subtracting the antiderivative evaluated at the lower limit from that evaluated at the upper limit: \[\left( \frac{250}{3} - \frac{75}{2} + 35 \right) - \left( -\frac{16}{3} - 6 - 14 \right)\].Simplify to obtain \( \frac{250}{3} - \frac{75}{2} + 35 + \frac{16}{3} + 6 + 14 \).Combine like terms and simplify the expression to arrive at the final result.

Key Concepts

Antiderivative

An antiderivative can be thought of as the reverse of taking a derivative. If you have a function and know its derivative, finding the antiderivative is a way to recreate the original function. This step is key in integral calculus. For instance, in our exercise, if we know the derivative of a function is \(2x^2 - 3x + 7\), we seek an antiderivative that will yield this expression when differentiated. The process involves reversing the common rules of differentiation. Here's how each term works in the example:

• For the term \(2x^2\), the antiderivative is \(\frac{2}{3}x^3\), because when differentiated, it results back in \(2x^2\).
• For \(-3x\), the antiderivative is \(-\frac{3}{2}x^2\), reversing it yields the original \(-3x\).
• The constant \(7\) becomes \(7x\), simply because the derivative of \(7x\) is \(7\).

These steps give us the complete antiderivative, which is crucial to solve our definite integral.

Polynomial Integration

Integrating a polynomial function is about applying the power rule of integration to each term of the polynomial separately. Polynomials are sums of terms that look like \(ax^n\). Integration here means increasing the power by one and dividing by the new power. Let's break it down:

• For each power term like \(x^n\), the antiderivative is \(\frac{x^{n+1}}{n+1}\). This formula helps us easily handle each part of the polynomial.
• Constant terms, such as 7 in our equation, convert into the term multiplied by \(x\). So, 7 becomes \(7x\).

Using this method, we derive the integrated form \(\frac{2}{3}x^3 - \frac{3}{2}x^2 + 7x\) from the original function \(2x^2 - 3x + 7\). This makes polynomial integration a straightforward application of rules, provided we carefully address each term.

Upper and Lower Limits Evaluation

When solving a definite integral, evaluating the antiderivative at the upper and lower limits gives the final solution. This process is often called the Fundamental Theorem of Calculus, which links differentiation and integration.Here’s what you do:

• First, plug the upper limit (in our case, 5) into the antiderivative we found, which yields a numerical value: \(\frac{2}{3}(125) - \frac{3}{2}(25) + 35\).
• Next, substitute the lower limit (-2) in the same function to get another value: \(\frac{2}{3}(-8) - \frac{3}{2}(4) - 14\).

Subtract the value obtained at the lower limit from that at the upper limit. This difference gives the area under the curve described by the polynomial \(2x^2 - 3x + 7\) between \(x = -2\) and \(x = 5\). Therefore, finding the specific numerical answer to a definite integral is about evaluating these limits—it’s about seeing how the function's total contribution changes across the interval.