Choose True if the statement is correct, otherwise False.1. Aleftendpoint Riemann sums will always result to underestimation of the area between the curve and thex-axis.2. Ifan integration process producesa constant number, it cannot be added with the constant of integration, and mustbe written as c + k where k is the other constant number.3. One cannot determine the definite integral if the integrand is discontinuous outside the limits of integration.4. Anegative definite integral for f (x)dx from a to b means that the total area bounded by a to bis below the x-axisand above the curve. Thus, any portion of the whole area taken within a and b is also below the x-axis and abovethe curve.5. All single-independent variable equations can somehow be differentiated. However, this is not the case forintegration since not all single-independent variable equations will always have an antiderivative thru standardmathematical operations.6. As long as an integrand is converging within the set limits of integration, one can always determine its estimateddefinite integral, no matter how complicated the integrand can be.7. By taking the average Riemann sums using the left and right endpoints, one is literally taking the area usingTrapezoidal rule.8. Ifa function is odd, its definite integral froma to b will always result to zero.9. The indefinite integral of a rational functionp(x)/q(x) with a discontinuity cannever be determined.10. If f(x)dx is to be integrated and u = f(x), then du = f'(x) for any function f(x). This is howu-substitutionworks.

As per bartleby guidelines for more than one question only first question to be answered please…

Answered: Choose True if the statement is…

Choose True if the statement is correct, otherwise False. 1. Aleftendpoint Riemann sums will always result to underestimation of the area between the curve and thex-axis.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Choose True if the statement is correct, otherwise False.
1. Aleftendpoint Riemann sums will always result to underestimation of the area between the curve and thex-axis.
2. Ifan integration process producesa constant number, it cannot be added with the constant of integration, and must
be written as c + k where k is the other constant number.
3. One cannot determine the definite integral if the integrand is discontinuous outside the limits of integration.
4. Anegative definite integral for f (x)dx from a to b means that the total area bounded by a to bis below the x-axis
and above the curve. Thus, any portion of the whole area taken within a and b is also below the x-axis and above
the curve.
5. All single-independent variable equations can somehow be differentiated. However, this is not the case for
integration since not all single-independent variable equations will always have an antiderivative thru standard
mathematical operations.
6. As long as an integrand is converging within the set limits of integration, one can always determine its estimated
definite integral, no matter how complicated the integrand can be.
7. By taking the average Riemann sums using the left and right endpoints, one is literally taking the area using
Trapezoidal rule.
8. Ifa function is odd, its definite integral froma to b will always result to zero.
9. The indefinite integral of a rational functionp(x)/q(x) with a discontinuity cannever be determined.
10. If f(x)dx is to be integrated and u = f(x), then du = f'(x) for any function f(x). This is howu-substitution
works.

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