Problem 46 - Calculus and its applications

Question

Evaluate. $$ \int_{1}^{8}(\sqrt[3]{x}-2) d x $$

Step-by-Step Solution

Answer

The evaluated integral is \(-\frac{11}{4}\).

Step 1: Identify the Components of the Integral

We identify the definite integral to evaluate: \[ \int_{1}^{8}(\sqrt[3]{x}-2) \, dx \]This consists of the function \(f(x) = \sqrt[3]{x} - 2\), and it is evaluated from \(x = 1\) to \(x = 8\).

Step 2: Rewrite the Integrand

The integrand \(\sqrt[3]{x}\) can be rewritten using exponent notation: \[f(x) = x^{\frac{1}{3}} - 2\] This makes it easier to integrate using the power rule.

Step 3: Apply the Power Rule for Integration

We integrate each term separately:1. Integrate \(x^{\frac{1}{3}}\):\[\int x^{\frac{1}{3}} \, dx = \frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} = \frac{x^{\frac{4}{3}}}{\frac{4}{3}} = \frac{3}{4}x^{\frac{4}{3}}\]2. Integrate \(-2\):\[\int (-2) \, dx = -2x\]Thus, the indefinite integral is:\[\frac{3}{4}x^{\frac{4}{3}} - 2x + C\]

Step 4: Evaluate the Definite Integral

We substitute the limits 1 and 8 into the evaluated antiderivative:\[\left[ \frac{3}{4}x^{\frac{4}{3}} - 2x \right]_{1}^{8}\]Calculate at the upper limit (\(x = 8\)):\[\frac{3}{4}(8)^{\frac{4}{3}} - 2(8) = \frac{3}{4}(16) - 16\]Calculate at the lower limit (\(x = 1\)):\[\frac{3}{4}(1)^{\frac{4}{3}} - 2(1) = \frac{3}{4} - 2 = \frac{-5}{4}\]

Step 5: Compute the Final Result

Subtract the lower limit from the upper limit:\[\left( \frac{3}{4}(16) - 16 \right) - \left( \frac{3}{4} - 2 \right) \]This simplifies to:\[12 - 16 - \left( \frac{3}{4} - 2 \right) = -4 + \frac{5}{4}\]Finding a common denominator gives:\[-\frac{16}{4} + \frac{5}{4} = -\frac{11}{4}\]

Step 6: State the Final Answer

The value of the definite integral is given by:\[\int_{1}^{8}(\sqrt[3]{x}-2) \, dx = -\frac{11}{4}\]

Key Concepts

Power Rule for Integration

Integration is a way to calculate the area under a curve. It's like finding the missing piece of a puzzle! One powerful technique to perform integration is the "Power Rule for Integration." This rule is especially handy when dealing with polynomials or terms of the form \(x^n\). The Power Rule states:

• If \(f(x) = x^n\), then the integral \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.

However, there are a few catches. This rule works only when \(n eq -1\) because when \(n = -1\), you're dealing with a logarithm (it's a special case!). In our original exercise, we rewrote \(\sqrt[3]{x}\) as \(x^{1/3}\) to easily apply the Power Rule. Remember, it's all about adding 1 to the exponent and dividing by this new exponent. After performing this step, you are left with the antiderivative, which can be further evaluated to find definite values.

Evaluating Limits

Evaluating limits in the context of definite integration is like focusing your camera lens to get a clear picture. We start with an antiderivative, a function we've integrated, and then plug in the limits, which are the numbers defining the interval of integration.First, substitute the upper limit (in this case \(x = 8\)) into the antiderivative. This gives us the area underneath the curve from the start to this point. Next, substitute the lower limit (\(x = 1\)) into the antiderivative. Finally, subtract the value obtained from the lower limit from the value at the upper limit.

• This subtraction represents only the area between the two numbers, effectively removing areas outside this boundary.

It's important to carry out each calculation accurately to ensure the correct net area is determined. This method ensures that no surplus area from outside the interest range is incorrectly included.

Antiderivative Computation

An antiderivative is simply a "backward" derivative. While derivatives involve finding the rate at which something changes, antiderivatives put the puzzle back together to find the original function. In our problem, once we've applied the Power Rule to both terms in the expression \(\sqrt[3]{x} - 2\), we get the antiderivative:

• \(\frac{3}{4}x^{4/3} \) for \(x^{1/3}\)
• -2x for -2

The complete antiderivative of the function \( \sqrt[3]{x} - 2\) becomes \(\frac{3}{4}x^{4/3} - 2x + C \). Here, \(C\) is often omitted in the context of definite integrals because the constant will cancel out when we evaluate the difference between the upper and lower limits.Understanding antiderivatives helps us track back from the rate of change to the accumulated quantity, shedding light on the graph or area that formed from changes over time.