Problem 211 - Calculus Volume 1

Question

Use basic integration formulas to compute the following antiderivatives. $$ \int_{0}^{\pi}(\sin x-\cos x) d x $$

Step-by-Step Solution

Answer

The result of the integral is 2.

Step 1: Identify the Function to Integrate

We need to find the antiderivative of the function \(\sin x - \cos x\). The definite integral is given from the limits \(0\) to \(\pi\).

Step 2: Find the Antiderivative

To find the antiderivative, recognize that the integral of \(\sin x\) is \(-\cos x\), and the integral of \(-\cos x\) is \(-\sin x\) plus a constant \(C\). Therefore, \[\int (\sin x - \cos x)dx = -\cos x + \sin x + C\].

Step 3: Evaluate the Definite Integral

Apply the Fundamental Theorem of Calculus to evaluate the definite integral from \(0\) to \(\pi\):\[\left[ -\cos x + \sin x \right]_0^\pi = \left( -\cos(\pi) + \sin (\pi) \right) - \left( -\cos(0) + \sin(0) \right)\].

Step 4: Simplify the Expression

Compute the trigonometric values: \(-\cos(\pi) = 1\), \(\sin(\pi) = 0\), \(-\cos(0) = -1\), and \(\sin(0) = 0\). Therefore, the expression simplifies to \[(1 + 0) - (-1 + 0) = 1 + 1 = 2\].

Key Concepts

Understanding Antiderivatives

Antiderivatives are the reverse process of derivatives. When you find the derivative of a function, you're determining the rate of change or slope of that function.
In contrast, finding an antiderivative means you're looking for a function whose derivative will give you the original function you began with.
This is often called "integration," and it's a key process in calculus.

• If you differentiate \(F(x)\) to get \(f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\).
• There can be multiple antiderivatives, differing by a constant. The constant part, often noted as \(C\), represents an infinite number of parallel functions shifted up or down.
• For example, if \(F(x) = -\cos x + \sin x + C\) is an antiderivative of \(\sin x - \cos x\), it means differentiating \(-\cos x + \sin x\) returns \(\sin x - \cos x\).

Evaluating Definite Integrals

Definite integrals are a core aspect of calculus, allowing you to calculate the accumulation of quantities, like area under a curve, between two points on a graph.
Unlike indefinite integrals, definite integrals provide a specific numerical value, representing change over an interval.

• To solve a definite integral, you'll need the antiderivative of the function (also called an indefinite integral), plus defined limits, such as from \(0\) to \(\pi\) in our example.
• After finding the antiderivative, you use the limits to calculate the net area: \([F(b) - F(a)]\), where \(b\) and \(a\) are the upper and lower limits, respectively.
• For the example \([ -\cos x + \sin x ]_0^\pi\), substitute \(\pi\) and \(0\) to get the specific value: \(2\).

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus ties together the concepts of derivatives and integrals, making calculus incredibly powerful.
It implies that differentiation and integration are inverse processes.

• Part one of the theorem states that if \(f\) is a continuous function on \([a, b]\), and \(F\) is an antiderivative of \(f\), then the integral from \([a, b]\) of \(f(x) dx\) is \([F(b) - F(a)]\).
• This shows that the process of finding an antiderivative (integration) and then differentiating (finding the rate of change) are like taking two steps forward and one step back.
• Understanding this relationship helps greatly in evaluating definite integrals, as seen in calculating the area under the curve \(\sin x - \cos x\) from \(0\) to \(\pi\).