Problem 33 - Essential Calculus

Question

Evaluate the definite integral. Use a graphing utility to confirm your result. $$ \int_{0}^{1 / 2} \arccos x d x $$

Step-by-Step Solution

Answer

The solution requires using integration by parts and substitution of the integral limits. The details of the solution are shown in the steps. The final result should be confirmed using a graphing utility, ensuring the area under the curve of the \( \arccos x \) function matches the calculated result.

Step 1: Setting up the integration by parts

Integration by parts follows the formula: \( \int u dv = uv - \int v du \). Set \( u = \arccos x \) and \( dv = dx \). Then differentiate and integrate to get \( du = -\frac{1}{\sqrt{1-x^2}}dx \) and \( v = x \).

Step 2: Apply the integration by parts formula

Now we can apply the integration by parts formula: \( \int u dv = uv - \int v du \). This gives us:\[ x\arccos(x) - \int -x / \sqrt{1-x^2} dx \]

Step 3: Resolve the remaining integral

The remaining integral \( \int -x/\sqrt{1-x^2} dx \) is a standard integral in the form: \( \int x\sqrt{a^2 - x^2} dx \) which results in \(-\frac{1}{3}*\sqrt{1-x^2}^3 \). Now we rewrite the original integral as:\[ x\arccos(x) + \frac{1}{3}\sqrt{1-x^2}^3 \].

Step 4: Evaluate the definite integral

Substitute the limits of the definite integral (0 and 1/2) into the expression \[ x\arccos(x) + \frac{1}{3}\sqrt{1-x^2}^3 \]to get the final result.

Step 5: Confirm the result graphically

Using a graphing calculator or software, sketch the graph of \( y = \arccos x \), and check that the area under the curve from \( x = 0 \) to \( x = 1/2 \) corresponds to the value obtained in Step 4. This will confirm the result.