Question

In Exercises 47-50, determine which value best approximates the definite integral. Make your selection on the basis of a sketch. $$\int_{0}^{4} \sqrt{x} d x$$ (a) 5 (b) -3 (c) 10 (d) 2 (e) 8

Short answer

According to the calculated area from the sketch, the best approximation of the definite integral \(\int_{0}^{4} \sqrt{x} dx\) from the given options would be (a) 5.

Step-by-step solution

Sketch the Function

Sketch the curve for the function \(y=\sqrt{x}\) in the interval [0,4]. It will be a curve starting from the origin (0,0) and moving upwards to the point (4,2).

Approximate the Area Under the Curve

The area under the curve of function from x=0 to x=4 can be approximated as a combination of simple geometric shapes (like rectangles or triangles). In this case, think of it as 2 triangles stacked to each other. The first one with base as 2 (from x=0 to x=2) and height as \(\sqrt{2}\), and the second one also with base as 2 (from x=2 to x=4) but height as 2 (as \(\sqrt{4} = 2\)). The area of a triangle is \(\frac{1}{2}\)base*height. Calculate the sum of the areas of these two triangles.

Choose the closest answer

Among the given options (a) 5, (b) -3, (c) 10, (d) 2, (e) 8. The calculated area from Step 2 should be compared, and the closest value should be chosen as the best approximate for the definite integral.