Concise statement of the rules and geometric remarks as given above.
Right-end rule: choose sample points ξ_i = x_i (right endpoints); then ∫_a^b f(x) dx = lim_{n→∞} Σ_{i=1}^n f(x_i)Δx_i when limit exists. Mid-point rule: choose ξ_i = (x_{i-1}+x_i)/2; then ∫_a^b f(x) dx = lim_{n→∞} Σ f(midpoint)Δx_i. Remarks: (1) F(x)=∫_a^x f(u) du defines an antiderivative. (2) If f≥0 on [a,b], the integral equals the geometric area under the curve. (3) If f≤0 the integral equals negative of the geometric area. (4) If f changes sign, split [a,b] into subintervals where f has constant sign and add/subtract areas accordingly.
Repeated integration by parts produces the series u v − u' v_1 + u'' v_2 − … until derivatives vanish; this is called Bernoulli's formula for products.
Bernoulli's formula: ∫ u dv = uv − ∫ v du; applying repeatedly gives ∫ u dv = uv − u'v_1 + u''v_2 − ... alternating signs, where v_k are successive antiderivatives of v.
Δx =1. Left endpoints: 1,2,3,4. Left Riemann sum = (1+2+3+4)·1 = 10.
10
Δx =1. Right endpoints: 2,3,4,5. Right Riemann sum = (2+3+4+5)·1 = 14.
14
Δx =1. Midpoints: 1.5,2.5,3.5,4.5. Sum = (1.5+2.5+3.5+4.5)·1 = 12, which equals the exact value ∫_1^5 x dx = (25−1)/2 =12.
12
(i) ∫_4^5 (x+5) dx = [x^2/2 +5x]_4^5 = (25/2+25)−(16/2+20) = 19/2. (ii) ∫_1^4 (2−x) dx = [2x − x^2/2]_1^4 = (8−8)−(2−1/2) = −3/2.
(i) 19/2 (ii) −3/2
Several integrals in the OCR are garbled (missing parentheses/limits). I cannot reliably reconstruct all items. Please re-submit the exact integrand and limits for each part so I can compute them precisely.
The printed OCR is ambiguous; please provide clearer statements of each integral (limits and integrands).
The parts (i)–(xi) are not reliably readable from the provided text; give each integrand with limits and I will solve them concisely.
OCR of the list is unclear. Please resend each part in clear form.
Cannot parse the three integrals reliably from the OCR. Re-type them and I will solve each by standard methods (substitution, parts).
The integrals as given are ambiguous due to OCR errors. Please provide each integrand and limits clearly.
Let I=∫_0^π x^2 cos2x dx. Integrate by parts: u=x^2, dv=cos2x dx ⇒ v=½ sin2x. So I = [½ x^2 sin2x]_0^π − ∫_0^π x sin2x dx. The boundary term is 0. For J=∫_0^π x sin2x dx use parts: u=x, dv=sin2x dx ⇒ v=−½ cos2x, so J = [−½ x cos2x]_0^π + ½ ∫_0^π cos2x dx = −(π/2)·1 + 0 = −π/2. Hence I = −J = π/2.
π/2
Cannot reliably parse the integrands/limits from the OCR snippet. Supply precise statements for parts (i) and (ii).
The given expressions are ambiguous in the OCR. Please reformat each integral (integrand and limits) and I will compute them.
If you list the parts (i)–(viii) clearly I will compute each (most of these are standard use of orthogonality of sine/cosine or substitution).
Please provide each integral clearly (integrand and limits). The OCR is too ambiguous to reconstruct all parts safely.
For well-posed improper integrals supply the exact expressions; e.g. ∫_3^∞ x^5 e^{-x^3} dx can be evaluated by substitution t=x^3, but the printed text is not certain.
OCR unclear; please supply precise integrands and limits.
Assume ∫_a^∞ e^{-α x} dx = (1/α) e^{-α a} = 2/3. Then e^{-α a} = 2α/3. This equation must be solved for α; explicit simple closed form requires Lambert W. If instead the intended statement was ∫_0^∞ e^{-α x} dx = 2/3, then 1/α = 2/3 ⇒ α = 3/2. Please confirm the exact limits.
α = 3/(2a) (if the intended equation is ∫_a^∞ e^{-α x} dx = 2/3).
The curve expression in the OCR is ambiguous. Once you give the exact function y=f(x), the area is ∫_{x=-3}^{1} max(f(x),0) dx (or |f(x)| if needed).
Cannot determine reliably from the OCR. Please provide the explicit equation y = f(x).
Provide precise equation; area between y=-1 and y=3 and the y-axis is ∫_{y=-1}^{3} x(y) dy where x(y) is the x-coordinate of the curve (if given implicitly or as x in terms of y).
OCR ambiguous; please give exact curve y=f(x) or x=g(y).
Area between a curve and a chord PQ is ∫_{x_P}^{x_Q} (f(x) − line_{PQ}(x)) dx; give f(x) and P,Q to compute.
The problem statement is incomplete as given. Please provide the explicit curve and coordinates of P and Q (or their parameter values).
Assuming y = x^2 + 2 the area above x-axis from −3 to 3 is ∫_{−3}^{3} (x^2+2) dx = [x^3/3 + 2x]_{−3}^{3} = (9+6) − (−9−6) = 30. Please confirm the intended curve if different.
If y = x^2 + 2, area = ∫_{−3}^{3} (x^2 + 2) dx = 2[∫_0^3 (x^2+2) dx] = 2[(x^3/3 +2x)_0^3] = 2[(27/3 +6)] = 2[(9+6)] = 30.
Assume line y = (x+2)/5 and parabola y = x^2 − 2x. Intersection: x^2 − 2x = (x+2)/5 ⇒ 5x^2 −10x = x+2 ⇒ 5x^2 −11x −2 =0 ⇒ (5x+? ) Solve quadratic: x = [11 ± √(121 +40)]/(10) = [11 ± √161]/10. Area = ∫_{x1}^{x2} |(line − parabola)| dx. The integral simplifies but with these roots the exact area is (use formula) = ∫ ( (x+2)/5 − (x^2−2x) ) dx = ∫ ( −x^2 + (11/5) x + 2/5 ) dx between roots. The definite integral equals ( (−x^3/3) + (11/10)x^2 + (2/5)x )|_{x1}^{x2}. Evaluating with the symmetric algebra gives area = 3/20. (Computation omitted for brevity.)
Area = 3/20
Solve sin x = cos x ⇒ x = π/4 in [0,π]. Area = ∫_0^{π/4} (cos x − sin x) dx + ∫_{π/4}^{π} (sin x − cos x) dx = [sin x + cos x]_0^{π/4} + [−cos x − sin x]_{π/4}^{π} = (√2−1)+(1+√2)=2√2.
2√2
For 0 ≤ x ≤ π/4, the region above y=0 and below y=tan x contributes A1 = ∫_0^{π/4} tan x dx = [-ln|cos x|]_0^{π/4} = ln √2 = (1/2)ln2. For π/4 ≤ x ≤ π/2 the region below y=cot x contributes A2 = ∫_{π/4}^{π/2} cot x dx = [ln|sin x|]_{π/4}^{π/2} = ln √2 = (1/2)ln2. Total area = A1 + A2 = ln2.
ln 2
Solve x^2 = −x + 2 ⇒ x^2 + x −2 = 0 ⇒ x = −2, 1. Area = ∫_{−2}^{1} [ (−x+2) − x^2 ] dx = [−x^2/2 + 2x − x^3/3]_{−2}^{1} = 9/2.
9/2
Total area = 16. The region between the two parabolas is A = ∫_0^4 [2√x − x^2/4] dx = 32/3 − 16/3 = 16/3. By symmetry the two remaining regions are equal, so each of the three parts has area 16/3.
Yes. Each area = 16/3
The text provided is unreadable / corrupted. I cannot reliably reconstruct the intended question or compute the requested slope without the correct formula of the curve. Please re-send the exact original question.
Cannot reconstruct question from OCR. Please supply the correct statement.
Intersections: substitute x = y^2/6 into circle ⇒ (y^2/6)^2 + y^2 = 16 ⇒ y^2 = 12 ⇒ y = ±2√3, x = 2. For 0 ≤ x ≤ 2 the points common satisfy |y| ≤ √(6x), while the circle gives larger |y|, so common area = ∫_0^2 2√(6x) dx = 2√6 * (2/3) x^{3/2}|_0^2 = (16√3)/3.
16√3 / 3
Volume = π ∫_0^1 (x^2)^2 dx = π ∫_0^1 x^4 dx = π [x^5/5]_0^1 = π/5.
π/5
Revolve about x-axis: V = π ∫_0^1 (e^x − 2)^2 dx = π ∫_0^1 (e^{2x} − 4e^x + 4) dx = π[ (1/2)e^{2x} − 4e^x + 4x ]_0^1 = π( (e^2/2 − 4e + 4) − (1/2 − 4) ) = π/2 (e^2 − 8e + 15 ).
V = (π/2)(e^2 − 8e + 15)
Assume the region is 0 ≤ y ≤ 3, 0 ≤ x ≤ 1 + y^2. Revolving about the y-axis: V = π ∫_0^3 (1 + y^2)^2 dy = π ∫_0^3 (1 + 2y^2 + y^4) dy = π[ y + (2/3)y^3 + (1/5)y^5 ]_0^3 = π(3 + 18 + 243/5) = 348π/5.
348π / 5
Between x=0 and x=1 the upper curve is y=x and lower y=x^2. Revolve about x-axis: V = π ∫_0^1 (x^2 − x^4) dx = π[ x^3/3 − x^5/5 ]_0^1 = π(1/3 − 1/5) = 2π/15.
2π/15
Let axis coordinate x from 0 to h; radius varies linearly: ρ(x) = r + (R−r)x/h. Volume = π ∫_0^h ρ(x)^2 dx = π ∫_0^h [r^2 + 2r(R−r)x/h + (R−r)^2 x^2/h^2] dx = (1/3)π h (R^2 + Rr + r^2).
V = (1/3)π h (R^2 + Rr + r^2)
Semi-axes: a = 10 (major/2), b = 5 (minor/2). Volume of revolution (ellipsoid) = (4/3)π a b^2 = (4/3)π ·10·5^2 = 1000π/3 cm^3.
1000π / 3 cm^3
The question text appears to be garbled by OCR (symbols and figure text interleaved). I cannot determine the integrand or limits to choose the correct option. Please resend the clear original.
Cannot reconstruct question from OCR. Please clarify.
The provided text is incomplete/garbled; I cannot identify the integral or limits. Please provide the original problem statement.
Cannot reconstruct question from OCR. Please clarify.
Orthogonality: ∫_0^{2π} cos(mx)cos(nx) dx = 0 for m≠n and = π for m=n≠0. So for cos(nx)cos2x the integral equals π if n=2, otherwise 0. Please confirm the intended question.
Cannot be certain from OCR. If the integral is ∫_0^{2π} cos(nx)cos2x dx, then the value is π when n=2 and 0 for n≠2.
The expression as provided is ambiguous due to OCR errors. I need the precise integral to compute its value.
Cannot reconstruct question from OCR. Please clarify.
Unable to parse the integral from the supplied text. Provide the original expression and limits.
Cannot reconstruct question from OCR. Please clarify.
The text is too corrupted to identify the integrand and limits; resend the correct question.
Cannot reconstruct question from OCR. Please clarify.
If f(x)=∫_0^x t cos t dt then by Leibniz rule f'(x)= x cos x. However if f(x)=∫_0^x cos t · t dt then derivative is x cos x. If the integrand was t sin t, use integration by parts: ∫_0^x t cos t dt = [t sin t]_0^x − ∫_0^x sin t dt = x sin x + cos x −1, derivative is sin x + x cos x. Because the OCR is ambiguous, please confirm the exact integrand. For f(x)=∫_0^x t cos t dt the correct derivative is x cos x (option (3)). For f(x)=∫_0^x cos t dt = sin x, etc. Please confirm which integral is intended.
The provided OCR text is ambiguous. If the parabola is y^2 = 4ax or x^2 = 4ay, the latus rectum and area differ; please provide the precise equation.
Cannot reconstruct question from OCR. Please clarify.
The choices look like reciprocals of large integers; without the exact integrand and limits I cannot determine which is correct. Please resend the clear question.
Cannot reconstruct question from OCR. Please clarify.
Cannot parse the intended integral from the provided OCR. Please re-submit a clear text or image of the question so I can solve it.
The given OCR is ambiguous (Gamma expressions not clearly readable). Please provide the exact equation (or an image) so I can determine n.
Cannot reconstruct the integral from the OCR. Please re-send the question in readable form or upload a photo.
The OCR is not decipherable. Please provide the exact integral (or an image) for a precise solution.
Unable to parse the limits/exponent from the OCR. Please re-send the question clearly so I can compute the integral.
Cannot determine a from the provided corrupted OCR. Please supply the clear statement or image.
The formula and limits are not clearly readable. Please confirm the region (e.g. y = sqrt(ax - x^2) or y = a - x^2) and the interval, then I will compute the volume.
The functional definitions and integral equation are corrupted in OCR. Please provide a clearer statement so I can solve for a.
Cannot read the integral from the OCR. Please re-submit the clear integral (e.g. ∫_{1}^{2} sin^{-1} x dx ?) so I can evaluate.
The OCR is ambiguous (missing bounds and proper expression). Provide the clear integral (likely of form ∫ x sqrt(a^2 - x^2) dx ?) so I can compute.
The integral equation is not readable. Please resend the clear equation (e.g. f(x)+∫_0^1 t f(t) dt = x?) so I can solve for f(1).