Integral Approximation and Error Analysis

Delta x = e^{(1.5)^2+3(1.5)+1}(0.3) = e^8.01(0.3) = 2973(0.3) = 891.90$$$$f(x_7) \Delta x = e^{(1.8)^2+3(1.8)+1}(0.3) = e^12.01(0.3) = 222000(0.3) = 66600$$$$
  • #1
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5
0

Homework Statement



Approximation of e^(x^2+3x+1) from 0 to 3 within .2 of the actual integral.


Homework Equations



Riemann Sum or Trapezoidal. (We haven't learned Taylor yet).

The Attempt at a Solution



Last n value i found was 148068 which gave me a delta x of 3/148068...im very confused.
 
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  • #2
Can someone help me solve this? I have the right answer for the integral (1953.2) but I'm not sure how to get there.So I am doing a right Riemann Sum: $$\sum_{i=1}^n f(x_i) \Delta x$$Where $$\Delta x = \frac{b-a}{n}$$ and $$f(x_i) = e^{x^2+3x+1}$$I'm given the interval [0, 3], so b = 3 and a = 0. Also, I'm told to approximate the integral within .2 of the actual integral. That means my goal is to make my calculation within .2 of 1953.2. So I can start off by choosing an arbitrary n value. Let's say n = 10. Then $$\Delta x = \frac{3-0}{10} = \frac{3}{10} = 0.3$$Now I just need to evaluate the function at each point on the interval and then multiply by the delta x. $$f(x_1) \Delta x = e^{0^2+3(0)+1}(0.3) = e^1(0.3) = 2.71828(0.3) = 0.815484$$$$f(x_2) \Delta x = e^{(0.3)^2+3(0.3)+1}(0.3) = e^1.51(0.3) = 4.52(0.3) = 1.3560$$$$f(x_3) \Delta x = e^{(0.6)^2+3(0.6)+1}(0.3) = e^2.51(0.3) = 12.2(0.3) = 3.66$$$$f(x_4) \Delta x = e^{(0.9)^2+3(0.9)+1}(0.3) = e^3.51(0.3) = 33.4(0.3) = 10.02$$$$f(x_5) \Delta x = e^{(1.2)^2+
 

Related to Integral Approximation and Error Analysis

What is integral approximation?

Integral approximation is a method used in mathematics and science to estimate the value of a definite integral. It involves dividing the interval of integration into smaller subintervals and using a function to approximate the area under the curve within each subinterval. The sum of these approximations gives an estimate of the total area under the curve.

How is integral approximation different from exact integration?

Exact integration involves finding the antiderivative of a function and evaluating it at the limits of integration. This gives the exact value of the definite integral. Integral approximation, on the other hand, uses numerical methods to estimate the value of the integral, which may not be exact but can be very close to the actual value.

What is the purpose of error analysis in integral approximation?

Error analysis is important in integral approximation because it allows us to quantify the accuracy of the estimated value of the integral. It involves calculating the difference between the estimated value and the actual value of the integral, and determining the sources of error such as the choice of approximation method, number of subintervals, and rounding errors.

What are some common methods of integral approximation?

Some common methods of integral approximation include the Trapezoidal Rule, Simpson's Rule, and the Midpoint Rule. These methods use different techniques to approximate the area under the curve and have varying levels of accuracy. Other methods include the Monte Carlo method and the Gaussian quadrature method.

How can we improve the accuracy of integral approximation?

There are several ways to improve the accuracy of integral approximation. One way is to increase the number of subintervals used in the approximation. Another way is to use a more accurate method, such as Simpson's Rule instead of the Trapezoidal Rule. Additionally, using a computer program or software can also improve the accuracy as it can handle larger numbers and perform calculations more precisely.

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