Use geometry and properties of integrals to evaluate the following definite integrals.
∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 . (Hint: Write the integral as sum of two integrals.)

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Step 1: Recognize that the integral can be split into two separate integrals using the property of linearity of integrals: ∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 = ∫₄⁰ 2𝓍 d𝓍 + ∫₄⁰ √(16―𝓍²) d𝓍.

Step 2: For the first integral, ∫₄⁰ 2𝓍 d𝓍, use the power rule for integration. The antiderivative of 2𝓍 is 𝓍². Evaluate this integral over the limits from 𝓍 = 0 to 𝓍 = 4.

Step 3: For the second integral, ∫₄⁰ √(16―𝓍²) d𝓍, recognize that √(16―𝓍²) represents the equation of a semicircle with radius 4 centered at the origin. The integral computes the area of the semicircle over the interval [0, 4].

Step 4: Use the formula for the area of a semicircle, A = (1/2)πr², where r is the radius. Here, r = 4. Compute the area of the semicircle corresponding to the interval [0, 4].

Step 5: Add the results of the two integrals together to obtain the final value of the definite integral ∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍.

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Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫[a,b] f(x) dx, where 'a' and 'b' are the limits of integration. The result of a definite integral is a numerical value that can be interpreted geometrically as the area between the curve and the x-axis from 'a' to 'b'.

Properties of Integrals

Properties of integrals, such as linearity, allow us to break down complex integrals into simpler parts. For instance, the integral of a sum can be expressed as the sum of integrals: ∫[a,b] (f(x) + g(x)) dx = ∫[a,b] f(x) dx + ∫[a,b] g(x) dx. This property is particularly useful for evaluating integrals that can be separated into more manageable components.

Geometric Interpretation of Integrals

The geometric interpretation of integrals involves visualizing the area under a curve. For the integral ∫[a,b] f(x) dx, the area can be calculated by considering the shape formed by the curve, the x-axis, and the vertical lines at 'a' and 'b'. Understanding this concept helps in evaluating integrals by recognizing geometric shapes, such as triangles or semicircles, that can simplify the calculation.

Use geometry and properties of integrals to evaluate the following definite integrals.∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 . (Hint: Write the integral as sum of two integrals.)