Question

In Exercises \(23-28,\) use areas to evaluate the integral. $$\int_{0}^{b} 4 x d x, \quad b>0$$

Short answer

The value of the integral \(\int_{0}^{b} 4x dx\) is \(2b^2\).

Step-by-step solution

Setup Integral

The integral given is: \(\int_{0}^{b} 4x dx\). We want to find the area under the curve of this function from 0 to b.

Find Antiderivative

The antiderivative of \(4x\) is \(2x^2\) because the power rule of integration states that the integral of \(x^n\) is \((x^{n+1}) / (n+1)\), here \(n = 1\). So in this case, the antiderivative is \(4x * x / 2 = 2x^2\).

Apply Limits of Integration

Now, apply the limits of integration to the antiderivative, \(2x^2\). Evaluate the antiderivative at the upper limit and then subtract the evaluation of the antiderivative at the lower limit. This gives us \((2b^2 - 2(0)^2)\).

Simplify the Solution

Simplify the expression to get the result. The answer is \(2b^2\).

Key concepts

Definite Integrals

Understanding definite integrals is pivotal in calculus, especially when it comes to calculating the area under a curve between two points on a graph. It is represented as an integral sign with lower and upper limits. For example, when we consider the integral \(\int_{0}^{b} 4x dx\), where \(b>0\), we are looking for the area under the curve of the function \(4x\) from \(x=0\) to \(x=b\).

Imagine shading the region under the curve from the starting point, \(0\), to any positive value of \(b\). The definite integral quantifies this shaded region, which in the context of a real-world scenario could represent anything from the total distance travelled to the quantity of a resource consumed over time.

Antiderivatives

An antiderivative, often called the indefinite integral, of a function is another function that, when differentiated, gives the original function. In our example, finding the antiderivative of \(4x\) involves determining what function, when differentiated, equals \(4x\). The antiderivative is typically denoted with a capital \(F\), and with respect to \(4x\), it can be represented as \(F(x) = 2x^2\), using the power rule of integration.

Once we've calculated the antiderivative, it gives us a general formula for the area under the curve, but to find the exact area up to a certain point—say \(b\)—we will need to use the limits of integration.

Power Rule of Integration

The power rule of integration is a straightforward method to integrate polynomials. When a function is given by \( x^n \) where \(n \)is any real number except \( -1\), the integral is given by \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration for indefinite integrals. In our exercise, applying the power rule to \(4x\), or \(x^1\), we increase the exponent by 1 to get \(x^2\) and then divide by the new exponent, yielding the antiderivative \(2x^2\).

This rule is essential as it provides a quick way to find antiderivatives of polynomial functions, a commonly occurring task in calculus.

Limits of Integration

The limits of integration define the interval over which we are integrating. In a definite integral, they correspond to the lower and upper bounds of the x-values. After finding the antiderivative, to calculate the definite integral, we evaluate the antiderivative at the upper limit and subtract the value of the antiderivative at the lower limit. This process is known as the Evaluation Theorem or the Fundamental Theorem of Calculus.

In the given exercise, the limits are from \(0\) to \(b\), so we evaluate \(2x^2\) at \(x=b\), getting \(2b^2\), and at \(x=0\), obtaining 0. Subtracting these gives us \(2b^2 - 0\), which simplifies to \(2b^2\), representing the exact area under the curve between \(x=0\) and \(x=b\).