Use the graph of the integrand to evaluate the integral. 4. int_(-1)^(1)(1+sqrt(1-x^(2)))dx

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Use the graph of the integrand to evaluate the integral. 4.  int_(-1)^(1)(1+sqrt(1-x^(2)))dx

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Identify Function & Limits: Identify the function to integrate and the limits of integration. The function to integrate is f(x) = 1 + sqrt(1 - x^2), and the limits of integration are from -1 to 1.Geometric Interpretation: Recognize the geometric interpretation of the integral. The function sqrt(1 - x^2) represents a semicircle with radius 1 centered at the origin. The integral of this function from -1 to 1 is the area of the semicircle. Adding 1 to the function raises the semicircle by 1 unit, creating a shape that includes the semicircle and a rectangle below it.Calculate Semicircle Area: Calculate the area of the semicircle. The area of a full circle with radius 1 is π(1)^2 = π. Since we only have a semicircle, its area is π/2.Calculate Rectangle Area: Calculate the area of the rectangle. The rectangle has a width of 2 (from -1 to 1) and a height of 1. Therefore, its area is 2 * 1 = 2.Find Total Area: Add the areas of the semicircle and the rectangle to find the total area under the curve. The total area under the curve from -1 to 1 is the area of the semicircle plus the area of the rectangle, which is π/2 + 2.Write Final Answer: Write the final answer. The definite integral of the function (1 + sqrt(1 - x^2)) from -1 to 1 is π/2 + 2.

Use the graph of the integrand to evaluate the integral. $\newline$ $\int_{-1}^{1}(1+\sqrt{1-x^{2}})\,dx$

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Use the graph of the integrand to evaluate the integral.\newline∫−11(1+1−x2) dx \int_{-1}^{1}(1+\sqrt{1-x^{2}})\,dx ∫−11​(1+1−x2​)dx

Use the graph of the integrand to evaluate the integral. $\newline$ $\int_{-1}^{1}(1+\sqrt{1-x^{2}})\,dx$