Problem 39 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the definite integral. \(\int_{0}^{1} \sqrt[3]{1+7 x} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{45}{28}\).

Step 1: Identify the integral

We are asked to evaluate the definite integral \(\int_{0}^{1} \sqrt[3]{1+7x} \, dx\). This involves finding the antiderivative of the function \(\sqrt[3]{1+7x}\) and evaluating it from \(x = 0\) to \(x = 1\).

Step 2: Choose a substitution

Let us use the substitution to simplify the integral. Choose \(u = 1 + 7x\), which implies \(du = 7 \, dx\) or \(dx = \frac{1}{7} du\).

Step 3: Change the limits of integration

When \(x = 0\), \(u = 1 + 7\times 0 = 1\). When \(x = 1\), \(u = 1 + 7\times 1 = 8\). So the limits of integration change from \(x=0\) to \(u=1\) and from \(x=1\) to \(u=8\).

Step 4: Substitute and simplify the integral

Substitute \(u\) and \(dx\) into the integral. The integral becomes \(\int_{1}^{8} \sqrt[3]{u} \cdot \frac{1}{7} \, du = \frac{1}{7} \int_{1}^{8} u^{1/3} \, du\).

Step 5: Evaluate the antiderivative

The antiderivative of \(u^{1/3}\) is \(\frac{u^{4/3}}{4/3} = \frac{3}{4} u^{4/3}\). Substitute this back into the integral: \[\frac{1}{7} \left[ \frac{3}{4} u^{4/3} \right]_{1}^{8}\].

Step 6: Compute definite integral using limits

Evaluate the antiderivative from \(u = 1\) to \(u = 8\):\[\frac{1}{7} \left[ \frac{3}{4} (8)^{4/3} - \frac{3}{4} (1)^{4/3} \right]\]. Since \(8^{4/3} = (2^3)^{4/3} = 2^4 = 16\), substitute back:\[\frac{1}{7} \times \frac{3}{4} \times (16 - 1) = \frac{1}{7} \times \frac{3}{4} \times 15 = \frac{45}{28}\].

Step 7: Final computation

After simplifying the expression, the value of the definite integral becomes \(\frac{45}{28}\).

Key Concepts

Substitution Method

The substitution method is a very handy technique used in calculus to simplify the integration process by transforming it into an easier form. This technique often involves replacing a complicated part of an integrand (the function being integrated) with a single variable, which is usually labeled as \(u\). By making this substitution, we aim to transform the integral into a simpler one.

In our exercise, we begin by substituting \(u = 1 + 7x\). Consequently, this substitution affects the differential and alters \(dx\) into \(du\), through the relationship derived from differentiating \(u\):

• \(du = 7 \, dx\)
• Thus, \(dx = \frac{1}{7} \, du\)

This choice of substitution simplifies the integral to a function of \(u\). By doing so, we simplify our work – turning more complex integrals into ones involving common functions that are easier to handle.

Integration Limits Change

When using substitution in definite integrals, it is crucial to change the limits of integration along with the expression. The limits set the boundaries for the definite integral, so when the variable changes, these boundaries must also be adjusted to maintain the integrity of the integration.

In our exercise:

• For the lower limit, \(x=0\), substitute into the expression \(u = 1 + 7x\) to obtain \(u=1\).
• For the upper limit, \(x=1\), using the same substitution yields \(u=8\).

By changing the limits from \([x=0, x=1]\) to \([u=1, u=8]\), you ensure that the whole process remains consistent and correct. Always remember to adjust the limits to match the new variable when solving definite integrals with substitution.

Antiderivative Calculation

Antiderivatives are essential for evaluating definite integrals. The antiderivative of a function represents a family of functions whose derivative gives the original function. In essence, it reverses the differentiation process.

For the function \(u^{1/3}\), finding the antiderivative involves integrating:

• The antiderivative is \(\frac{u^{4/3}}{4/3}\).
• This simplifies to \(\frac{3}{4} u^{4/3}\) by multiplying through by the reciprocal of \(4/3\).

This step is key to solving the integral, as it allows you to then evaluate it over the new limits. The skill to find antiderivatives is a cornerstone of mastering calculus, especially for solving applied mathematical problems.

Cubed Root Function Integration

Integrating cubed root functions primarily involves recognizing the function's structure and using both substitution and antiderivative skills.

In our problem, we deal with \(\sqrt[3]{1+7x}\), which translates to \((1+7x)^{1/3}\). A substitution transforms this into a simpler integral function of \(u\) — \(u^{1/3}\).

By computing the definite integral of \(u^{1/3}\) over the new limits \([1,8]\), we find:

• First, substitute the antiderivative, resulting in \(\frac{3}{4} \left[ u^{4/3} \right]_{1}^{8}\).
• This calculation gives \(\frac{3}{4} (8^{4/3} - 1^{4/3})\).
• Since \(8^{4/3} = 16\), compute \(16 - 1 = 15\). Multiply by \(\frac{3}{4}\), and finally, adjust for \(\frac{1}{7}\) outside.

This method turns an intimidating function into a tangible series of logical steps, leading to the final solution.