Problem 67 - Calculus and its applications

Question

Solve for \(K\), given that $$ \int_{1}^{2}\left[\left(3 x^{2}+5 x\right)-(3 x+K)\right] d x=6 $$

Step-by-Step Solution

Answer

The value of \(K\) is 4.

Step 1: Simplify the Integrand

First, simplify the function inside the integral. Given is \[\int_{1}^{2}\left[\left(3 x^{2}+5 x\right)-(3 x+K)\right] d x = 6\]Simplify it by combining like terms:\[\int_{1}^{2} \left(3x^2 + 5x - 3x - K\right) dx = \int_{1}^{2} \left(3x^2 + 2x - K\right) dx\]

Step 2: Integrate the Simplified Function

Now, find the antiderivative of the simplified function:\[\int \left(3x^2 + 2x - K\right) dx\]The antiderivatives are:\[\int 3x^2 \; dx = x^3, \quad \int 2x \; dx = x^2, \quad \int K \; dx = Kx\]Thus, the antiderivative becomes:\[x^3 + x^2 - Kx + C\] where \(C\) is a constant of integration.

Step 3: Evaluate the Definite Integral

Evaluate the antiderivative from 1 to 2:\[\left[ x^3 + x^2 - Kx \right]_1^2\]Substitute the upper and lower limits:\[(2^3 + 2^2 - 2K) - (1^3 + 1^2 - K)\]Simplify:\[(8 + 4 - 2K) - (1 + 1 - K)= (12 - 2K) - (2 - K)= 12 - 2K - 2 + K= 10 - K\]

Step 4: Solve the Equation

Set the evaluated integral equal to 6, as given by the problem:\[10 - K = 6\]Rearrange and solve for \(K\):\[K = 10 - 6 = 4\]

Key Concepts

Definite Integral

A definite integral is used to calculate the accumulation of quantities, like areas under curves. It is expressed as \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration, representing the interval over which the function \(f(x)\) is integrated.
To understand a definite integral conceptually, think of it as the total area between the graph of \(f(x)\) and the \(x\)-axis, over the interval from \(a\) to \(b\).

• Upper and Lower Limits: The numbers \(a\) and \(b\) are known as the lower and upper limits of integration, respectively.
• Net Area: Areas above the \(x\)-axis are positive, while those below are negative, contributing to the "net" integral value.

In the exercise, the goal was to determine \(K\) such that the definite integral of the expression evaluates to a specific value, in this case, 6.

Antiderivative

The antiderivative, also known as the indefinite integral, is the reverse of differentiation. When you integrate a function, you are essentially finding another function whose derivative matches the original function. This process involves adding a constant of integration \(C\), as derivatives wipe out constants during differentiation.
For example, if the derivative of \(F(x)\) yields \(f(x)\), then \(F(x) + C\) is the antiderivative of \(f(x)\).
In the problem, the antiderivative of \(3x^2 + 2x - K\) is calculated as \(x^3 + x^2 - Kx + C\). This expression is crucial to evaluate the definite integral between specific limits.

Integration Techniques

Integration techniques are strategies used to find the antiderivative of functions. Basic polynomial functions, such as in our problem, are integrated by applying the power rule: increasing the power by 1 and dividing by the new exponent

• For \(3x^2\): Increase 2 by 1 to get 3, resulting in \(x^3\).
• For \(2x\): Increase power 1 by 1 to get 2, resulting in \(x^2\).
• The term involving the constant \(K\) is simply integrated as \(Kx\).

These straightforward techniques can be applied directly to obtain the antiderivatives, simplifying further calculations.
If the function were more complex, other techniques such as substitution or integration by parts might be necessary.

Simplifying Expressions

Simplifying expressions is an essential step in solving problems involving algebraic and calculus operations. It involves combining like terms and reducing expressions to a simpler form, making further calculations more manageable.
In the given exercise, the expression inside the integral was initially \((3x^2 + 5x) - (3x + K)\). It was subsequently simplified by:

• Combining the \( + 5x\) and \( - 3x\) terms to obtain \(+2x\);
• Reducing the expression to \(3x^2 + 2x - K\).

This simplification made it easier to integrate, focusing on fewer terms and directly applying the known integration techniques.
Simplified expressions not only save time but also reduce the likelihood of mistakes in later steps.