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Problem 3 Medium Difficulty

Estimate $ \displaystyle \int_0^1 \cos (x^2)\ dx $ using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with $ n = 4 $. From a graph of the integrand, decide whether your answers are underestimates or overestimates. What can you conclude about the true value of the integral?

Answer

a.$\approx 0.895759$
b.0.9089

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Video Transcript

there are problems. Three, uh before. So you need to know. Zero two times. One over four. Just two times over too. And two times three or four. Uh, sorry. After your four and one times one bye Seems interval is one or four. You should be ready by two. Yes, there's a to here. Right? Use a calculator. You can find out there t is equal to zero point age. Nice aches. And to use a meat point rule, you need to know have one or eight and thus 3.3 over eight. And us, uh five. All right. And seven or eight. Seems the lance often throw is one over four. So they should multiply when the war for right and you can find on is equal to zero point ni oh Ni uh, from a broth, I formulated profits in Mathematica. Here scenes. The graph is concave and is decreasing. Right, So we can know that and is greater than I busies concave. And it's greater than team right? Here's a graph. He is a raft for cribs. Oi! And there's a space here and the cost. Yet I'm is greater than eyes because the function is concave, right? So what we can know? We can conclude that eyes between to appoint a United States and to appoint night all night.