Let A=int_(0)^(1)cos xdx. We estimate A using the L,R, and T approximations with n=100 subintervals. Which is true? (A) L

Question

Let  A=int_(0)^(1)cos xdx. We estimate  A using the  L,R, and  T approximations with  n=100 subintervals. Which is true? (A)  L < A < T < R (B)  L < T < A < R (C)  R < A < T < L (D)  R < T < A < L

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Understand Approximations: Let's first understand what L, R, and T approximations are. The Left (L) approximation sums up the values of the function at the left endpoints of the subintervals, the Right (R) approximation sums up the values of the function at the right endpoints, and the Trapezoidal (T) approximation averages the left and right approximations. Since cos(x) is decreasing on the interval [0, 1], we know that the Left approximation will overestimate and the Right approximation will underestimate the actual integral value. The Trapezoidal approximation, being an average of L and R, should lie between them. Calculate Left Approximation: We can calculate the Left approximation (L) by summing the values of cos(x) at the left endpoints of the subintervals. Since cos(x) is decreasing on [0, 1], L will be an overestimate. Calculate Right Approximation: Similarly, we calculate the Right approximation (R) by summing the values of cos(x) at the right endpoints of the subintervals. Since cos(x) is decreasing on [0, 1], R will be an underestimate. Calculate Trapezoidal Approximation: The Trapezoidal approximation (T) is the average of L and R. Since L is an overestimate and R is an underestimate, T should be closer to the actual value A than either L or R. Compare Approximations: Now, we can compare the approximations. Since cos(x) is decreasing on [0, 1], we have R < A < L. The Trapezoidal approximation T should be between L and R, so we have R < T < L. Combining these, we get R < T < A < L.

Let $A=\int_{0}^{1}\cos x\,dx$ . We estimate $A$ using the $L$ , $R$ , and $\newline$ $T$ approximations with $n=100$ subintervals. Which is true? $\newline$ (A) L < A < T < R $\newline$ (B) L < T < A < R $\newline$ (C) R < A < T < L $\newline$ (D) R < T < A < L

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Let A=∫01cos⁡x dxA=\int_{0}^{1}\cos x\,dxA=∫01​cosxdx. We estimate AAA using the LLL, RRR, and \newlineTTT approximations with n=100n=100n=100 subintervals. Which is true?\newline(A) L < A < T < R\newline(B) L < T < A < R\newline(C) R < A < T < L\newline(D) R < T < A < L

Let $A=\int_{0}^{1}\cos x\,dx$ . We estimate $A$ using the $L$ , $R$ , and $\newline$ $T$ approximations with $n=100$ subintervals. Which is true? $\newline$ (A) L < A < T < R $\newline$ (B) L < T < A < R $\newline$ (C) R < A < T < L $\newline$ (D) R < T < A < L