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.)

Verified step by step guidance

1. 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.
2. 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.
3. 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].
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].
5. Step 5: Add the results of the two integrals together to obtain the final value of the definite integral ∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍.

Key Concepts

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.