Question

Evaluate the following definite integrals. Find \(k\) if \(\int_{0}^{2}\left(x^{3}+k\right) d x=10\).

Short answer

Answer: The value of the unknown constant k is 3.

Step-by-step solution

Evaluate the definite integral without the unknown constant k

To evaluate the integral \(\int_{0}^{2}(x^3) dx\), we'll use the power rule for integration: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\). So, in this case, we have: \(\int_{0}^{2} x^3 dx = \frac{x^{4}}{4} \Big|_0^{2}\)

Evaluate the definite integral including the constant k

Now, we evaluate the integral with the unknown constant \(k\): \(\int_{0}^{2} (x^3 + k) dx = \int_{0}^{2} x^3 dx + k\int_{0}^{2} dx\) Evaluating each part, we get: \(\frac{x^{4}}{4} \Big|_0^{2} + k(x) \Big|_0^{2}\) Plugging in the limits of integration: \(\left(\frac{2^4}{4} - \frac{0^4}{4} \right) + k(2-0)\) This simplifies to: \(\frac{16}{4} + 2k = 4 + 2k\)

Solve for k

Now, we need to set the evaluated integral equal to 10 and solve for \(k\): \(4 + 2k = 10\) Subtract 4 from both sides: \(2k = 6\) Divide by 2: \(k = 3\) So, the value of \(k\) is 3.

Key concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, often used to find the exact area under a curve between two points. Unlike indefinite integrals, a definite integral is evaluated over a specific interval \(a, b\), producing a numerical value rather than a function with a constant \(C\). In our example, the definite integral \(\int_{0}^{2}(x^{3}+k) dx\) calculates the area under the curve \(x^3 + k\) from \(x = 0\) to \(x = 2\).
The notation \(\int_{a}^{b} f(x) \, dx\) can be read as the integral of \(f(x)\) from \(a\) to \(b\). Once evaluated, this definite integral yields an actual number that represents the accumulated sum of area under the curve, inclusive of all the mathematical transformations within the specified interval.

Power Rule for Integration

The power rule for integration is a straightforward method to find antiderivatives of functions in the form of \(x^n\). The rule states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\). This formula helps simplify the integration process by providing an algebraic expression for powers of \(x\).
In our exercise, we used the power rule to integrate \(x^3\):

• Set \(n = 3\). This gives \(\frac{x^{4}}{4}\).
• Evaluate at the bounds as \(x\) changes from 0 to 2 using the fundamental theorem of calculus.

Such rules make it easier to handle polynomial integrals, reducing them to simple expressions. This allows us to focus on applying limits to find precise regions of areas, crucial for solving the definite integral at hand.

Evaluating Integrals

Evaluating integrals involves calculating the actual numerical value represented by the integral after its function has been integrated, and the limits of integration are applied. In this problem, after finding the indefinite integral of \(x^3\), we proceed with evaluating it from 0 to 2.
Using the integration results:

• Substitute the upper limit \(2\) into \(\frac{x^4}{4}\) to get \(\frac{16}{4} = 4\).
• Substitute the lower limit \(0\) into \(\frac{x^4}{4}\) which equals 0, so it subtracts without altering the result.
• Adding the influence of \(k\), evaluate the integral \(\int_{0}^{2} k dx = k(x) \big|_0^2 = 2k\).

Combine these results accordingly, leading to the expression \(4 + 2k\), as per our calculation step. Evaluating integrals efficiently allows us to determine final outcomes that convey the total influence of the function over the specified range.

Solving Equations

Solving equations routinely emerges during integral evaluations to complete the solution. Particularly in this exercise, after finding the expression \(4 + 2k\), we equate this to 10, as per the problem's condition. This involves standard algebraic methods to isolate and solve for the unknown variable \(k\).
Steps include:

• Separate the known value from the expression by subtracting 4: \(2k = 6\).
• Isolate \(k\) by dividing both sides by 2, resulting in \(k = 3\).

Solving such equations is a crucial back-calculation in applied calculus problems, enabling us to turn mathematical expressions into specific numerical solutions. These calculations often help verify the integrity or result of an initial problem while synthesizing integration and algebraic manipulation.