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Use the graph of the integrand to evaluate the integral.
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Q. Use the graph of the integrand to evaluate the integral.
- Identify Function & Limits: Identify the function to integrate and the limits of integration. The function to integrate is , and the limits of integration are from to .
- Geometric Interpretation: Recognize the geometric interpretation of the integral. The function represents a semicircle with radius centered at the origin. The integral of this function from to is the area of the semicircle. Adding to the function raises the semicircle by 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 is . Since we only have a semicircle, its area is .
- Calculate Rectangle Area: Calculate the area of the rectangle.The rectangle has a width of (from to ) and a height of . Therefore, its area is .
- 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 to is the area of the semicircle plus the area of the rectangle, which is .
- Write Final Answer: Write the final answer.The definite integral of the function from to is .
