Question

Approximate Integration and Improper Integrals (a) (3 points) Example: Suppose we are approximating an integral $\int_a^b f(x) dx$ using the trapezoidal approximation $T_n$. The diagrams below show that if $f''(x)$ is positive (the graph of $f(x)$ is concave up), then $T_n$ is always an overestimate and if $f''(x)$ is negative (the graph of $f(x)$ is concave down), then $T_n$ is always an underestimate. Now suppose we are approximating an integral $\int_a^b f(x) dx$ using the right endpoint ap- proximation $R_n$. Copy and complete the following sentences and briefly explain why you are correct (diagrams might be a helpful). If $f'(x)$ is ______, then $R_n$ is always an overestimate. If $f'(x)$ is ______, then $R_n$ is always an underestimate. (b) (7 points) Evaluate $\int_0^1 x^3 \ln(x^8) dx$

          Approximate Integration and Improper Integrals
(a) (3 points) Example: Suppose we are approximating an integral $\int_a^b f(x) dx$ using the
trapezoidal approximation $T_n$. The diagrams below show that if $f''(x)$ is positive (the graph
of $f(x)$ is concave up), then $T_n$ is always an overestimate and if $f''(x)$ is negative (the graph
of $f(x)$ is concave down), then $T_n$ is always an underestimate.
Now suppose we are approximating an integral $\int_a^b f(x) dx$ using the right endpoint ap-
proximation $R_n$. Copy and complete the following sentences and briefly explain why you are
correct (diagrams might be a helpful).
If $f'(x)$ is ______, then $R_n$ is always an overestimate.
If $f'(x)$ is ______, then $R_n$ is always an underestimate.
(b) (7 points) Evaluate $\int_0^1 x^3 \ln(x^8) dx$

Approximate Integration and Improper Integrals
(a) (3 points) Example: Suppose we are approximating an integral ^b f(x) dx using the
trapezoidal approximation Tn. The diagrams below show that if f”(x) is positive (the graph
of f(x) is concave up), then Tn is always an overestimate and if f”(x) is negative (the graph
of f(x) is concave down), then Tn is always an underestimate.
Now suppose we are approximating an integral ^b f(x) dx using the right endpoint ap-
proximation Rn. Copy and complete the following sentences and briefly explain why you are
correct (diagrams might be a helpful).
If f'(x) is , then Rn is always an overestimate.
If f'(x) is , then Rn is always an underestimate.
(b) (7 points) Evaluate ∫0^1 x^3 ln(x^8) dx

Added by Thomas E.

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