Use geometry and properties of integrals to evaluate the following definite integrals.

∫₀⁴ √(8𝓍―𝓍²) d𝓍 . (Hint: Complete the square .)

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Step 1: Recognize that the integral ∫₀⁴ √(8𝓍 - 𝓍²) d𝓍 represents the area under the curve of the function √(8𝓍 - 𝓍²) from x = 0 to x = 4. To simplify the integrand, complete the square for the expression inside the square root: 8𝓍 - 𝓍².

Step 2: Rewrite 8𝓍 - 𝓍² as - (𝓍² - 8𝓍). Factor out the negative sign and complete the square for the quadratic expression inside the parentheses: 𝓍² - 8𝓍 = (𝓍 - 4)² - 16. Thus, 8𝓍 - 𝓍² becomes -((𝓍 - 4)² - 16) = 16 - (𝓍 - 4)².

Step 3: Substitute the completed square form into the integral. The integral becomes ∫₀⁴ √(16 - (𝓍 - 4)²) d𝓍. Notice that this represents the equation of a semicircle with radius 4 centered at x = 4.

Step 4: Use the geometric interpretation of the integral. The function √(16 - (𝓍 - 4)²) describes the upper half of a circle with radius 4. The area under this curve from x = 0 to x = 4 corresponds to a quarter-circle of radius 4.

Step 5: Calculate the area of the quarter-circle using the formula for the area of a circle, A = πr². Since this is a quarter-circle, the area is (1/4)πr², where r = 4. Substitute r = 4 into the formula to find the area.

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Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫[a,b] f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The value of a definite integral can be interpreted geometrically as the net area between the function f(x) and the x-axis over the interval [a, b]. Understanding how to evaluate definite integrals is crucial for solving problems involving area and accumulation.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is particularly useful in integration, as it simplifies the integrand into a form that is easier to integrate. For example, the expression ax² + bx + c can be rewritten as a(x - h)² + k, where (h, k) is the vertex of the parabola. This technique is essential for evaluating integrals involving square roots of quadratic expressions.

Geometric Interpretation of Integrals

The geometric interpretation of integrals involves visualizing the area under a curve as a sum of infinitesimally small rectangles. This concept is foundational in calculus, as it connects the algebraic process of integration with geometric ideas. By understanding how to represent functions graphically, students can better grasp the significance of definite integrals in calculating areas, volumes, and other physical quantities, making it easier to approach problems involving integrals.

Use geometry and properties of integrals to evaluate the following definite integrals. ∫₀⁴ √(8𝓍―𝓍²) d𝓍 . (Hint: Complete the square .)