Question

Evaluate the integral \(\int_{0}^{10} \sqrt{10 x-x^{2}} d x\) by completing the square and applying appropriate formulas from geometry.

Short answer

#Short Answer The value of the definite integral \(\int_{0}^{10}\sqrt{10x-x^2}dx\) is \(\frac{25}{2}\pi\), which represents the area of a semicircle with a radius of 5.

Step-by-step solution

Completing the square

In the given expression, \(10x - x^2\), we want to rewrite it in the form of \((x-a)^2 + b\) where a and b are constants. We can achieve this by the method of completing the square. \(10x - x^2 = -x^2+10x = -(x^2-10x)\) Now, to complete the square, we need to add and subtract half of the coefficient of x squared, which is \(\frac{10}{2} = 5\): \(-(x^2-10x) = -(x^2 - 10x + 5^2 - 5^2) = -(x^2 - 10x + 5^2) + 5^2 = - (x-5)^2 + 5^2\)

Rewrite the integral

Now that we have completed the square, we can rewrite the integral in the form of geometrical shape. \(\int_{0}^{10}\sqrt{10x-x^2}dx = \int_{0}^{10}\sqrt{-(x-5)^2 + 5^2}dx\) The integrand represents the equation of a semicircle: \(\sqrt{-(x-5)^2 + 5^2}\) The equation corresponds to a semicircle centered on \((5,0)\) with a radius of 5. Since the integral has limits in the interval \([0,10]\), we're calculating the positive half of the semicircle.

Apply geometry to calculate the area of the semicircle

Now that we've recognized the integral as representing the area of a semicircle, we can apply the geometry formula for the area of a circle to find the area of the semicircle and thus the value of the integral. The area of a circle with radius \(r\) is given by: \(A_{circle} = \pi r^2\) Since we have a semicircle, we can take half of this formula for the area: \(A_{semicircle} = \frac{1}{2}\pi r^2\) Our radius in this case is \(r=5\), so the area of the semicircle is: \(A_{semicircle} = \frac{1}{2}\pi (5)^2 = \frac{25}{2}\pi\)

Conclude the result

We determined that the integral represents the area of a semicircle with a radius of 5. Using the area formula for a semicircle, we found that the area is \(\frac{25}{2}\pi\). Therefore, the value of the integral \(\int_{0}^{10}\sqrt{10x-x^2}dx\) is: \(\int_{0}^{10}\sqrt{10x-x^2}dx = \frac{25}{2}\pi\)

Key concepts

Completing the Square

The technique known as completing the square is a cornerstone of algebra that provides a methodological approach to manipulate quadratic expressions. This method transforms a quadratic expression of the form ax^2 + bx + c into a perfect square trinomial (x-h)^2 + k, revealing its maximum or minimum and making it easier to integrate or solve equations.

The process involves dividing the coefficient of the x term by 2, squaring the result, and adding as well as subtracting it within the expression. For the integral in question, \(10x - x^2\), this process yields \( -(x^2 - 10x + 5^2) + 25\), which rearranges to \( - (x-5)^2 + 25\). This new form is especially useful for the given problem because it directly leads to interpreting the integral as the area of a semicircle.

Geometrical Interpretation of Integrals

Integrals in calculus can often be understood and evaluated using geometry, especially when dealing with functions that represent physical shapes. The interpretation of an integral as the area under a curve is one such instance, and it's a significant concept that aids in visual learning and comprehension.

When we complete the square for the function within the integral \(\sqrt{10x - x^2}\), we are able to recognize it as the equation for a semicircle. This geometrical insight allows for applying area formulas from geometry rather than calculating the integral analytically. The visualization of the integrand as part of a circle's circumference clarifies the relationship between the algebraic expression and the physical space it occupies, fostering a deeper understanding of the integral calculus at hand.

Area of a Semicircle

The area of a semicircle, which is half of a circle, is one of the geometric formulas that we often use in calculus. This area can be determined by using the formula for the area of a circle, \(\pi r^2\), and then dividing by two since a semicircle is precisely half of the full circle. Hence, the formula for the area of a semicircle becomes \(\frac{1}{2} \pi r^2\).

In the given exercise, we discover that the integral represents the area of a semicircle with a known radius (\(r = 5\)). Consequently, we apply this formula to find that the area, and thus the value of the integral, is \(\frac{1}{2} \pi (5)^2 = \frac{25}{2} \pi\). It's essential to relate the algebraic integral to the geometric shape as it not only eases the computation but also enhances spatial reasoning skills in interpreting mathematical concepts.