DAGE-696 We have to find the value of the integral ∫x² dx and evaluate it using different methods.
### Step 1: Understanding the Integral
The integral ∫x² dx represents the area under the curve y = x² between a certain range of x values. To find the value of this integral, we'll need to evaluate it over a specific interval, which hasn't been specified. For the sake of this exercise, let’s assume the integral is to be evaluated between 0 and 1.
### Step 2: Evaluating the Integral Using Antiderivatives
First, let's use the antiderivative method to evaluate the integral.
**Antiderivative:**
The antiderivative of x² is found by using the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration.
Applying this rule to x²:
∫x² dx = (x³ / 3) + C
**Evaluating the Integral Between 0 and 1:**
We can now evaluate the integral between 0 and 1 by plugging in these values:
∫x² dx = (1³ / 3) – (0³ / 3) = (1 / 3) – (0 / 3) = 1 / 3
So, the integral evaluates to 1 / 3.
### Step 3: Evaluating the Integral by Riemann Sum
Another way to evaluate integrals is to use the Riemann sum. The Riemann sum approximates the area under a curve by dividing it into small rectangles and summing their areas.
**Step 3.1: Dividing the Interval**
Let's divide the interval [0, 1] into n equal subintervals. Each subinterval will have a length of h = 1 / n.
**Step 3.2: Choosing Points for Each Subinterval**
We'll choose the right endpoints of each subinterval as the points to evaluate the function. So, for the kth subinterval, the point will be xₖ = k / n.
**Step 3.3: Calculating the Riemann Sum**
The Riemann sum is then formed by summing up the areas of these rectangles:
Rₙ = ∑(k=1 to n) (xₖ)² × h
= ∑(k=1 to n) (k / n)² × (1 / n)
= ∑(k=1 to n) (k² / n³)
= (1 / n³) ∑(k=1 to) k²
**Step 3.4: Taking the Limit as n→∞**
To evaluate the integral, we take the limit of the Riemann sum as n approaches infinity:
∫x² dx = lim(n→∞) Rₙ
= lim(n→∞) (1 / n³) ∑(k=1 to) k²
Now, we need to find the value of ∑(k=1 to) k².
**Step 3.5: Finding the Sum of Squares**
A known formula for the sum of squares is:
∑(k=1 to n) k² = (n(n + 1)(2n + 1)) / 6
Plugging this into our Riemann sum:
Rₙ = (1 / n³) × (n(n + 1)(2n + 1)) / 6
= ((n(n + 1)(2n + 1)) / (6n³)
Now, taking the limit:
∫x² dx = lim(n→∞) ((n(n + 1)(2n + 1)) / (6n³)
= lim(n→∞) ((n² + n)(2n + 1)) / (6n³)
= lim(n→∞) (2n³ + n² + 2n² + n) / (6n³)
= lim(n→∞) (2n³ + 3n² + n) / (6n³)
= lim(n→∞) (2 / 6) + (3 / 6n) + (1 / 6n²)
= (1 / 3) + 0 + 0
= 1 / 3
Again, the integral evaluates to 1 /。
### Step 4: Evaluating Using Integration by Parts
Let's also try evaluating the integral using integration by parts.
**Integration by Parts:**
Integration by parts states that:
∫u dv = uv - ∫v du
Let's choose u = x and dv = x dx.
Then, du = dx and v = (x² / 2)
Applying the formula:
∫x² dx = x * (x² / 2) - ∫(x² / 2) dx
= (x³ / 2) - (∫x² / 2 dx)
But we are trying to find ∫x² dx, so this method doesn't help us directly. Let's try a different approach.
**Alternative:**
Let's choose u = x² and dv = dx
Then du = 2x dx and v = x
Applying the formula:
∫x² dx = x * x - ∫x * 2x dx
= x² - 2∫x² dx
This leads us back to the original integral, so it doesn't help.
**Another Approach:**
Let's choose u = x and dv = 2x dx
Then du = dx and v = x²
Applying the formula:
∫x² dx = x * x² - ∫x² * dx
= x³ - ∫x² dx
Again, we find ourselves in the same situation.
Therefore, integration by parts doesn't seem to be an effective method for this integral.
### Step 5: Evaluating Using Substitution Method
Let's try using the substitution method to evaluate the integral.
**Substitution:**
Let u = x²
Then du = 2x dx => xdx = du / 2
This doesn't directly help us since there's no x in the integrand.
Hence, the substitution method seems ineffective here.
### Step 6: Conclusion
We've evaluated the integral using two primary methods: the antiderivative approach and Riemann sum. Both methods have confirmed that the integral evaluates to 1 / 3. Integration and differentiation are inverse processes, so we've used both methods successfully to find the value of the integral.
**Final Answer:**
∫x² dx = x³ / 3 + C
5月 23日 2013年
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