in exercises 1 - 6, (a) use the trapezoidal rule with n = 4 to approximate the value of the integral. (b)…

in exercises 1 - 6, (a) use the trapezoidal rule with n = 4 to approximate the value of the integral. (b) use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. finally, (c) find the integral’s exact value to check your answer.\n1. $int_{0}^{2}x dx$\n2. $int_{0}^{2}x^{2}dx$\n3. $int_{0}^{2}x^{3}dx$\n4. $int_{1}^{2}\frac{1}{x}dx$\n5. $int_{0}^{4}sqrt{x}dx$\n6. $int_{0}^{pi}sin x dx$

in exercises 1 - 6, (a) use the trapezoidal rule with n = 4 to approximate the value of the integral. (b) use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. finally, (c) find the integral’s exact value to check your answer.\n1. $int_{0}^{2}x dx$\n2. $int_{0}^{2}x^{2}dx$\n3. $int_{0}^{2}x^{3}dx$\n4. $int_{1}^{2}\frac{1}{x}dx$\n5. $int_{0}^{4}sqrt{x}dx$\n6. $int_{0}^{pi}sin x dx$

Answer

  1. (a) Using the Trapezoidal Rule for (\int_{0}^{2}x dx):
    • The Trapezoidal - Rule formula is (T_n=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+\cdots + 2f(x_{n - 1})+f(x_n)]), where (\Delta x=\frac{b - a}{n}), (a = 0), (b = 2), and (n = 4).
    • First, calculate (\Delta x):
      • (\Delta x=\frac{2 - 0}{4}=\frac{1}{2}).
      • The sub - intervals are ([0,\frac{1}{2}],[\frac{1}{2},1],[1,\frac{3}{2}],[\frac{3}{2},2]), and (x_0 = 0,x_1=\frac{1}{2},x_2 = 1,x_3=\frac{3}{2},x_4 = 2).
      • (f(x)=x), so (f(x_0)=0), (f(x_1)=\frac{1}{2}), (f(x_2)=1), (f(x_3)=\frac{3}{2}), (f(x_4)=2).
      • (T_4=\frac{\frac{1}{2}}{2}[0 + 2\times\frac{1}{2}+2\times1+2\times\frac{3}{2}+2])
        • (=\frac{1}{4}(0 + 1+2 + 3+2))
        • (=\frac{8}{4}=2).
  2. (b) Analyzing concavity:
    • The second - derivative of (y = f(x)=x) is (y''=0). Since (y'' = 0), the Trapezoidal Rule gives the exact value (neither an over - estimate nor an under - estimate).
  3. (c) Finding the exact value:
    • Using the power rule for integration (\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), for (\int_{0}^{2}x dx=\left[\frac{x^{2}}{2}\right]_{0}^{2}).
      • (\frac{2^{2}}{2}-\frac{0^{2}}{2}=2).

Answer:

(a) (2) (b) Neither over - estimate nor under - estimate (c) (2)