?Draw the graph of \(f(x)=\sin \left(\frac{1}{2} x^{2}\right)\) in the viewing rectangle

Chapter 7, Problem 4

(choose chapter or problem)

Draw the graph of \(f(x)=\sin \left(\frac{1}{2} x^{2}\right)\) in the viewing rectangle \([0, 1]\) by \([0, 0.5]\) and let \(\mid=\int_{0}^{1} f(x) \ d x\).

(a) Use the graph to decide whether \(\mathrm{L}_{2}, \ \mathrm{R}_{2}, \ \mathrm{M}_{2}, \text { and } \mathrm{T}_{2}\) underestimate or overestimate \(I\).
(b) For any value of \(n\), list the numbers \(\mathrm{L}_{n}, \ \mathrm{R}_{n}, \ \mathrm{M}_{\mathrm{n}}, \ \mathrm{T}_{\mathrm{n}}, \text { and } \mathrm{I}\) in increasing order.
(c) Compute \(\mathrm{L}_{5}, \ \mathrm{R}_{5}, \ \mathrm{M}_{5},  \text { and } \mathrm{T}_{5}\). From the graph, which do you think gives the best estimate of \(I\)?

Equation Transcription:

f(x) = sin(

[0, 1]

[0, 0.5]

I = ∫ f(x)dx

L2, R2, M2, and T2

I

n

Ln, Rn, Mn, Tn, and I

L5, R5, M5, and T5

I

Text Transcription:

f(x)=sin(1/2x^2)

[0, 1]

[0, 0.5]

I=integral _0 ^1 f(x)dx

L_2, R_2, M_2, and T_2

I

n

L_n, R_n, M_n, T_n, and I

L_5, R_5, M_5, and T_5

I