Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the definite integral. $$ \int_{2}^{5}(-3 x+4) d x $$

Step-by-Step Solution

Answer

The value of the definite integral is -19.5.

Step 1: Find the antiderivative

The antiderivative of the function \(-3x + 4\) can be found by taking the antiderivative of each term separately. The antiderivative of \(-3x\) is \(-1.5x^2\) and the antiderivative of \(4\) is \(4x\). Combining these, the antiderivative of \(-3x + 4\) is \(-1.5x^2 + 4x\).

Step 2: Evaluate the antiderivative at the upper limit

Evaluate the antiderivative \(-1.5x^2 + 4x\) at the upper limit of \(5\). This gives \(-1.5*(5)^2 + 4*(5) = -37.5 + 20 = -17.5\).

Step 3: Evaluate the antiderivative at the lower limit

Evaluate the antiderivative \(-1.5x^2 + 4x\) at the lower limit of \(2\). This gives \(-1.5*(2)^2 + 4*(2) = -6 + 8 = 2\).

Step 4: Apply the Fundamental Theorem of Calculus

Subtract the antiderivative evaluated at the lower limit from the antiderivative evaluated at the upper limit, \(-17.5 - 2 = -19.5\).

Key Concepts

Antiderivatives

Antiderivatives are a crucial concept when dealing with integrals. An antiderivative of a function is essentially the reverse of taking a derivative. In simpler terms, it is finding a function that would yield the original function when differentiated.

To find the antiderivative of the polynomial function \(-3x + 4\) in our original exercise, we look at each term separately.

• The antiderivative of \(-3x\) is found using the power rule: increase the power of \(x\) by one and divide by the new power, leading to \(-1.5x^2\).
• For the constant \(4\), the antiderivative is simply \(4x\), as differentiating \(4x\) gives back \(4\).

Combining these, the antiderivative we get for \(-3x + 4\) is \(-1.5x^2 + 4x\). This forms the basis for calculating the definite integral.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of differentiation and integration. It comes in two parts:

1. The first part asserts that integration and differentiation are inverse processes.2. The second part provides a way to calculate definite integrals using antiderivatives.

• For any function \(f(x)\), if \(F(x)\) is its antiderivative, the definite integral from \(a\) to \(b\) can be found using: \(F(b) - F(a)\).

In our exercise, this theorem was applied after finding the antiderivative \(-1.5x^2 + 4x\). We evaluate this at the upper and lower limits (5 and 2, respectively) and subtract, giving \(-17.5 - 2 = -19.5\). By summarizing functions rather than lists of numbers, calculus provides a powerful tool, showing not just numeric solutions but insights into their continuous behavior.

Evaluating Integrals

Evaluating a definite integral involves several key steps, seamlessly combining our understanding of antiderivatives and the Fundamental Theorem of Calculus.

Once we have the antiderivative, here’s how we proceed:

• First, plug the upper limit into the antiderivative function to calculate \(F(b)\).
• Next, plug the lower limit into the same antiderivative function to calculate \(F(a)\).
• Finally, subtract \(F(a)\) from \(F(b)\) to get the definite integral.

For example, with the antiderivative \(-1.5x^2 + 4x\); after evaluating at the limits the results are \(-17.5\) (upper) and \(2\) (lower). The difference, \(-17.5 - 2\), gives us the integral of \(-19.5\). This process, while seemingly straightforward, encapsulates powerful calculus principles, enabling us to tackle more complex and subtle problems.