Approximating Area Under a Curve

What is the approximate area of the region, using 4 subintervals and heights using left values?

• 7.5

• 7.75

• 10

• 11.5

Estimate the area of the region using two intervals and heights using right values. Then estimate the area using four intervals and heights using right values.

What is the difference between the two estimates?

• 0

• 2.5

• 5.75

• 8

A Riemann Sum uses rectangles to

• approximate the area under a curve. The more rectangles, the better the approximation.

• approximate the area under a curve. The more rectangles, the worse the approximation.

• approximate the area under a curve. The less rectangles, the better the approximation.

• none of these

Which of the following is true about Riemann sums?

• They can approximate the area under a curve by summing the areas of rectangles.

• They can only be used with continuous functions.

• They provide an exact value for the area under a curve.

• They are more accurate than the Trapezoidal Rule and Simpson's Rule for all functions.

It is an approximate area of a region, obtained by adding up the areas of multiple simplified slices of the region.

• Riemann sum

• definite integral

• summation notation

• sum of a series

• 17

• 53

• 15

• 44

• 15.5

• 12.15

• 13.25

• 11.5

Which of the following best describes the error estimation in numerical integration?

• The sum of the absolute values of the derivatives of the function being integrated

• The difference between the exact value of the integral and its numerical approximation

• The maximum value of the function over the interval of integration

• The number of subintervals used in the approximation method

• 3.5

• 3.0

• 2.5

• 2.75

Which of the following statements is true regarding the use of numerical integration methods?

• Numerical integration methods can only approximate integrals of polynomial functions.

• Numerical integration methods are always less accurate than finding the exact integral.

• Numerical integration methods are useful for approximating integrals when the antiderivative of the function is difficult or impossible to find.

• Numerical integration methods are unnecessary when the antiderivative of the function can be easily found.