Final answer:
To approximate the area under the curve of the function f(x)=x^2+1 from a=0 to b=1 using 5 rectangles, calculate a Riemann sum using the heights of rectangles at right-endpoints. For the exact area, evaluate the definite integral of f(x) from a to b using the Fundamental Theorem of Calculus.
Step-by-step explanation:
To approximate the area under the curve of the function f(x) = x^2 + 1 from a=0 to b=1 using 5 rectangles, we would calculate a Riemann sum. Since we are splitting the interval into 5 equal parts, each rectangle will have a width of 0.2 (since (1-0)/5 = 0.2). To use the right-endpoint for our calculation, the heights of the rectangles at each interval will be f(0.2), f(0.4), f(0.6), f(0.8), and f(1). We then add the areas of these rectangles to get our Riemann sum approximation of the area under the curve.
For the exact area, we utilize the Fundamental Theorem of Calculus, and evaluate the definite integral of f(x) from a to b. The definite integral of f(x) is the antiderivative of f(x) evaluated at b minus the evaluation at a. In mathematical terms, this is ∫ab(x^2 + 1) dx, which equals to [x^3/3 + x] evaluated from 0 to 1.
By plugging in the values of a and b, we find the exact area under the curve, which provides the probability in continuous probability functions, where PROBABILITY = AREA.